I suspect that the phrase "more bang for the buck" is an instance in which an actual or imagined outlier obscures the overwhelmingly more likely source of a phrase's popularization.
Analyzing the outliers from 1935 and 1940
As I noted last year in a comment beneath the posted question, the instance of the phrase that is supposedly from a 1935 article in Milestones in Analytical Chemistry (cited in the posted question) is extremely unlikely to be from 1935, given that, as the blurb for the book says,
Over the past 65 years, the ACS journal Analytical Chemistry has published seminal papers on almost every development in the discipline. This volume presents a history of the field of analytical chemistry through original research papers published in the Journal from 1935 to the present.
The 1940 instance mentioned in the posted question is noted (but not linked to) by The Phrase Finder in its examination of the phrase "bang for the buck":
The first citation of the phrase in print that I can find is an advert in Metals and Plastics Publications, 1940. No advertiser would use a colloquial expression in an advert unless it was understandable to his audience, so I expect there are earlier citations yet to be found.
Unfortunately, I can find no specific information about the wording of this particular advertisement. A search of the 1940 edition of Metal Finishing: Preparation, Electroplating, Coating (volume 38 of the journal from Metals and Plastics Publications) at Hathi Trust for the words "bang AND buck" did not yield any page matches, meaning that (according to Hathi Trust's internal search engine) there are no individual pages in the issues of the magazine for 1940 that contain both the word bang and the word buck.
Since Metal Finishing was published for more than a century (its last publisher, Elsevier, announced that publication of the journal would cease as of December 31, 2013), there may be a typo in the year of publication or an OCR error leading to a misidentification of the correct year—in which case the problem isn't that there is no such advertisement, but that the advetisement appeared in a year other than 1940. But my preliminary conclusion from this brief side-investigation is that the 1940 advertisement that The Phrase Finder cites is at best unconfirmed and at worst spurious.
The avalanche of instances from 1953–1955
Even if it were confirmed as a 1940 occurrence, the cited instance of "more bang for the buck" is not obviously the direct source of phrase's subsequent popularity. That's because the normal wording of the phrase during the early 1950s, when a flurry of instances did occur, was "more bang for a buck."
Google Books and Elephind searches for "more bang for the/a/my/your/our/its/his/her/their/every buck and for "more bang for the bucks" yielded 16 matches from the period from 1953 through 1955. Here they are, in roughly chronological order.
From Stewart Alsop, "Strategic Decision," in the San Bernardino [California] Sun (December 23, 1953):
The joint chiefs accordingly went back and re-estimated their requirements, on the theory that atomic fire power can be substituted for man power and conventional fire power. This "bigger bang for a buck" theory was not, to be sure, the only factor in the proposed cutbacks.
Indeed there are plenty of informed officials and officers who privately believe that the requirements of the budget came before the requirements of national security when the original "new look" was revised downward. They believe that the "more bang for a buck" theory is an excuse for the cutbacks rather than the real reason, and that the "buck" came first by an easy margin, with the "bang" a poor second.
From "More Bang for a Buck," in the Madera [California] Tribune (January 19, 1954):
Most everyone will sympathize with the motive involved in the preferential contract idea, but there are many who question its practicality. California's Senator Knowland, for one, is openly dubious of the merits of the proposal. Senator Knowland contends that the idea behind defense contracts is to get the country the most defense for the fewest dollars. In the Department of Defense, it is called "more bang for a buck."
From "Move to Slash U.S. Defense Costs," in the [Hobart, Tasmania] Mercury (January 21, 1954):
The Eisenhower Administration feels that the development and stockpiling of new weapons, particularly nuclear weapons, now makes possible economies in both manpower and conventional weapons without any substantial loss of safety.
This is the concept jocularly known at the Defence Department as that of a "bigger bang for a buck."
The Defence Department has let it be known that it intends to slash Army manpower by 18 p.c. during 1954-55.
From "Don Iddon's New York Diary: Pneumatic Drill Is The Spring Song," in the [Adelaide, South Australia] Advertiser (March 26, 1954):
Eisenhower's "new look" foreign and defence policy is getting a certain amount of attention, but there have been so many confusing statements about "massive retaliatory blows" that voters don't which way to dodge.
The slang definition of the "new look" is "more bang for a buck"— a buck being a dollar.
From U.S. Atomic Energy Commission, In the matter of J. Robert Oppenheimer: transcript of hearing before Personnel Security Board (April 12, 1954—May 6, 1954) [combined snippets]:
There had been, I think, some thought about weapons development after I left Los Alamos. There was one meeting which I could not attend on the thermonuclear program, and there were lots of things left over from the wartime to get people interested in making better weapons, better here meaning a whole lot of things. lt means obviously getting more bang for a buck. It means more economy in the use of fissionable material. It means getting weapons which give you the maximum versatility in the kind of delivery system we have, so you don't have to use very big bombers and so on.
From Brassey's Annual (1954) [combined snippets]:
The crux of the argument is that the political leaders have been dishonest in suggesting that atomic weapons can lead to an economy in manpower, that the nuclear battle requires more, not less men than the conventional battle since it necessitates a much higher degree of decentralisation, and that fewer or cheaper misapprehensions have been foisted on to the taxpayer than the slogan "a bigger bang for a buck."
The organisation of the new weight accorded to air power as publicised by the civilians in the Administration in the form of "more strength for less money" or "a bigger bang for a buck" has been a source of considerable unhappiness to senior officers, whether airborne, landborne, seaborne or chairborne.
From Machinery, volume 61, issues 1–6 (1954):
As a result, Washington is taking another look" at our defenses. Such things as the "new look" and a "bigger bang for a buck" are not discussed any more. Instead there is a new concept of defense which might develop into the idea of a Fortress North America.
It may mean that the heads of the Department of Defense are finding it easier to yield to the "more and bigger defense facilities" argument than to hold out for the "bigger bang for a buck." Whatever the reason, the word has gone out that there must be increased appropriations for defense in the half-dozen years ahead, starting as of right now.
From Congressional Record, Proceedings of the Debates of the ... Congress, volume 100, part 12 (1954) [combined snippets]:
The 1954 cut was sold by the sloganmakers as giving more "bang for a buck," and being a "New Look," based upon "instant, massive retaliatory" power. When the facts were closely analyzed, it was found that we were getting less bangs for fewer bucks; that our power of retaliation was decreased rather than increased.
From The Commonweal, volume 61 (1954):
If we are to be militarily strong and yet not "spend ourselves into bankruptcy," obviously the answer is good management which will give us more defense for less money, "more bang for a buck." [page 33]
Instead, there was a headlong flight into the folly of a "bigger bang for a buck" and Secretary Dulles's misleading hints that the threat of the weapons could achieve diplomatic and political results in current international relations which the weapons themselves could never achieve and for which they should never be employed.
From The Nation, volume 178 (1954) [combined snippets]:
There is, for instance, the effort to make drastic cuts in defense expenditures and still show the national defense is somehow stronger instead of weaker. (It is referred to locally, with regrettable lack of reverence, as the "more bang for a buck" program.)
From Newsweek, volume 44, issues 1–13 (1954):
Ridgway's argument that [the] U.S. couldn't fulfill its military commitments with only seventeen divisions [was] answered by beefing up the Army [with] atomic weapons. Proponents of the [New] Look insisted that these would give [the] Army "a bigger bang for a buck."
From "Hobby with a Bang—Big Guns in Miniature," in Popular Mechanics, volume 102 (September 1954):
Cost of firing even the largest model is estimated by Herd at about one-half cent a shot.
"And that's, to use a current. Army slogan," Herd says, "a lot of bang for a buck!"
From "Bolling Calls for Dispersal of Industries," in the San Bernardino, [California] Sun (February 8, 1955):
Boiling said the greatest obstacle to overcome is the argument of "economy firsters" that present defense programs are adequate.
"We had better forget 'more bang for a buck' and choose 'more peace for a price'," he said.
From Facts Forum News, volume 4, issue 9 (1955):
The heart and core of that defense force must be continuing and expanding emphasis upon research and development. This includes both basic and applied research. Here the present administration flounders and vacillates. Nor will slogans suffice—"bigger bang for a buck" does not mean greater defense.
From Oil, Chemical & Atomic Union News, volumes 11–14 (1955[?]) [text not shown in snippet window]:
It was intended to avoid reductions in the Marine Corps, planned in line with the Administration's "bigger bang for a buck" theory. Final decision on how the money will be spent rests with the Administration, however.
From National Aeronautics, volumes 31–35 (1955[?]) [combined snippets]:
And they probably will answer the Wilson "more bang for a buck" talk by providing more bucks for defense, and particularly for air power. This will be good news for the aviation industry and it might provide the Democrats with good campaign material in 1956.
As you can see, all 16 of these instances contain the phrase "bang for a buck," although they disagree about whether the lead-in to that phrase should be "bigger," "more," or "a lot of."
Early variants from the later 1950s
The first instance of "bang for the buck is from The Air Reservist (1957):
Air Guard — "More Bang for the Buck"
Another variant—"bang for our buck"—debuts in Industrial Marketing, volume 44, issues 1–6 (1959) [combined snippets]:
With relatively small outlays and many promises the Communists are getting a much greater run for their ruble than we are getting a bang for our buck.
And also in 1959 we see the first match for "bang for its buck," from Barron's National Business and Financial Weekly, volume 39 (1959) [combined snippets]:
The Office of Naval Research, one of the country's outstanding champions of pure science is vulnerable to the budget-pairing knife. The Pentagon, as everyone knows, wants more bang for its buck.
Although these instances account for a very small proportion of the total number of instances of "bang for ... buck" that reached print during the 1950s, they include the form that ultimately overcame "bang for a buck" and became the colloquial standard of the 1960s: "bang for the buck."
It would be difficult to find a case in which the popularization of a phrase has a clearer point of origin than "bang for the buck." The U.S. military adopted it informally in late 1953 (in the form of "more bang for a buck") and publicized it persistently for the next several years as part of the defense department's New Look strategy at the close of the Korean War.
The central premise of the New Look strategy was that nuclear weapons were a cheaper and better deterrent to military aggression than a large conventional military force would be. Literally, the "bang" was the sound of a detonating nuclear warhead; figuratively it was the promise of superior security at a lower dollar price.
The claims for earlier instances of "more bang for the buck" in 1935 or 1940 are dubious although not impossible. However, given that the wording "bang for a buck" utterly dominated the examples in print during the period 1953–1955, and given that the expression became extremely popular during this same period and never really went away (although the popular wording of the phrase definitely shifted to "bang for the buck" during the 1960s), the question of whether isolated occurrences of the phrase may also have popped up 18 or 13 years before the U.S. military slogan appeared seems effectively to be beside the point.