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On the tangential issue of whether "let's don't" is syntactically correct, I observe that there are very few print occurrences of the uncontracted form "let us do not" in the sense of "let us not do [something]." One of the few is this instance from "Big Crop Means Ruin for Farmers," in the [Cloverport, Kentucky] Breckenridge News (April 7, 1909):

On the tangential issue of whether "let's don't" is syntactically correct, I observe that there are very few print occurrences of the uncontracted form "let us do not" in the sense of "let us not do [something]." One of the few is this instance from "Big Crop Means Ruin for Farmers," in the [Cloverport, Kentucky] Breckenridge News (April 7, 1909):

Anita Auer, "Let's not, let's don't, and don't let's in British and American English,"in Marianne Hundt, ed., Late Modern English Syntax (2014) offers a detailed discussion of these forms and their relative popularity over time. Only part of the article is visible in the Google Books link, however.

The decline of let us not (which is roughly one-sixth as frequent as let's not in COHA results from the first decade of the twenty-first century, according to Auer's Figure 3.1 chart) is evident across the nineteenth century and continued across the twentieth century, aside from a significant uptick in usage during the 1940s. Despite this trajectory, don't let's didn't surpass it until about 1920.

A scholarly assessment of 'let's not,' let's don't,' and 'don't let's'

Anita Auer, "Let's not, let's don't, and don't let's in British and American English,"in Marianne Hundt, ed., Late Modern English Syntax (2014) offers a detailed discussion of these forms and their relative popularity over time. Only part of the article is visible in the Google Books link, however.

The decline of let us not (which is roughly one-sixth as frequent as let's not in COHA results from the first decade of the twenty-first century, according to Auer's Figure 3.1 chart) is evident across the nineteenth century and continued across the twentieth century, aside from a significant uptick in usage during the 1940s. Despite this trajectory, let's not didn't surpass it until about 1920.

A side excursion to the wilds of 'let's don't'

On the tangential issue of whether "let's don't" is syntactically correct, I observe that there are very few print occurrences of the uncontracted form "let us do not" in the sense of "let us not do [something]." One of the few is this instance from "Big Crop Means Ruin for Farmers," in the [Cloverport, Kentucky] Breckenridge News (April 7, 1909):

Let us do not deceive ourselves and look for Providence to help us out of a bad fix if we with our eyes open get into one, for Providence helps those who help themselves.

The effect is somewhat ludicrous, as if the writer had started with "Let's don't" and then spelled out the contracted words for maximum solemnity—which may very well be what happened. Something similar occurs when people submit letters to the editor in which they replace familiar contracted forms of expression such as "Don't we have ..." with would-be formal wording such as "Do not we have ..."

A critique of the contracted form "let's don't" appears in Frank Colby, "Don't Take My Word for It" in the San Bernardino [California] Sun (August 21, 1941):

LET'S

Wrong: "Let's don't drive so fast." No. Let's is the contraction for LET US. Don't is the contraction for DO NOT. "Let us do not, therefore, is at once seen to be absurd. Better say: Let's not drive so fast.

A few years later, the same syndicated columnist returns to the topic in the same newspaper, although in the interim he has updated its standing title from "Don't Take My Word for It" to "Take My Word for It"—an interesting change in imperative. From Frank Colby, "Take My Word for It," in the San Bernardino [California] Sun (January 6, 1944):

Q. Why doesn't someone tell _____ (a radio M. C.) not to say "Let's you and I"?

A. Such errors as "let's you and I go, let's us not go, let's don't go" are quite common. When it is pointed out that "let's" is simply the contraction for "let us," one doesn't need to be a grammarian (and who is these days?) to see the absurdity of saying, "let us you and I go; let us us not go; let us do not go." The first redundant; the second is an impossible locution; the third is grammatically incorrect as it contains two verbs—"let" and "do." Correct: Let us (or let's) go let us (or let's) not get.

Scarcely 15 months after that, Colby is back for a third bite of the apple. From "Take My Word for It" in the San Bernardino [California] Sun (May 8, 1946):

I cannot think of another term in English so often used as the contraction "let's." First of all, "let's" is simply short for "let us," the literal meaning of which is, "permit us to ..."

But in common usage, "let's" has taken on a rather vague meaning that is equivalent to "suppose we ...?" or "what do you say if we ...?" And, in writing the apostrophe is often omitted, as, "lets go for a walk."

The most frequent misuses are "Let's don't," "let's us don't," and "don't let's us." In fact, we are so accustomed to such expressions as "Let's don't argue about it; let's us don't quarrel; don't let's us fail to go," that few of us realize that what we are actually saying is: "Let us do not argue about it; let us us do not quarrel; do not let us us fail to go."

Colby's presentation here treats the contracted form of "let us do not" as no more acceptable than the contracted form of "let us us do not"—although he surely sees that the latter introduces a second level of syntactical faultiness. However, the beginning of his third article acknowledges the definitional basis for using "let's don't" in a syntactically coherent way: people understand the "let's" component of the phrase to mean not "permit us to..." but "suppose that we...?" And if that is the intended meaning of "let's" in this construction, it seems to me that we can reasonably treat "let's don't" as an idiomatic set phrase meaning not "let us do not" but "suppose that we do not." This is not to say that "let's don't" is in good standing in formal speech or writing, interchangeably with "let's not"; but it is to say that there is a logic to its use.

'Do let's don't'

Anita Auer, "Let's not, let's don't, and don't let's in British and American English,"in Marianne Hundt, Late Modern English Syntax (2014) has offers a detailed discussion of these forms and their relative popularity over time. (Only part of the article is bible in the Google Books link, however.

One of Auer's observations seems especially relevant to the posted question:

These selected comments [from reviewers at goodreads.com] indicate that let's don't is considered to be bad syntax and a grammatical error that readers do not expect to be used by highly educated characters. The reviewer in [comment] 2b points out that one 'should say "Let's not do that"' rather than 'Let's don't do that'. In fact, it appears that in Present-Day English (PDE) let's don't is not the only alternative to let's not, as we also increasingly come across constructions with don't let's. All three forms can be found both in British English (BrE) and American English (AmE), albeit in different frequency rates. For PDE, Algeo (2006: 30) observes that 'don't let's is 7 times more frequent in British than in American, and let's don't is 4.5 times more frequent than don't let's in American'.

Also relevant to the question is Figure 3.1 in Auer's article (page 50 of Hundt's collection), which indicates that, whatever the relative popularity of let's not, let's don't, and don't let's may have been in the nineteenth century, the print occurrence of all three were dwarfed by instances of let us not—at least in the United States (the chart in Figure 3.1 draws on data from the Corpus of Historical American English).

The decline of let us not (which is roughly one-sixth as frequent as let's not in COHA results from the first decade of the twenty-first century, according to Auer's Figure 3.1 chart) is evident across the nineteenth century and continued across the twentieth century, aside from a significant uptick in usage during the 1940s. Despite this trajectory, don't let's didn't surpass it until about 1920.

By the early 2000s, let us not was not much more common than let's don't and don't let's in COHA results. Still, omitting let us not from consideration can lead to a very skewed impression of what wording was most common during the 188s and early 1900s.

Least common of all is do let's don't, which appears chiefly (and perhaps exclusively) in comic or parodic settings. From Harry De Puy, "Bull in the Night," in Hemingway Review (1981), cited in James McKelly, "For Whom the Bull Flows: Hemingway in Parody," in Louis Budd & Edwin Cady, eds., On Humor: The Best from 'American Literature' (1992), who cites this an example of parodic language mocking Hemingway's "'Anglicization' of England's English":

"Oh, give a chap a break."

"I say, don't, what?

"What?"

"What?"

"What?"

"I said, I say, don't, what? What?"

"Will I not!"

I say, do don't."

"What?"

"I say, do let's don't. Do. Don't."

"Oh, rather."

Anita Auer, "Let's not, let's don't, and don't let's in British and American English,"in Marianne Hundt, ed., Late Modern English Syntax (2014) offers a detailed discussion of these forms and their relative popularity over time. Only part of the article is visible in the Google Books link, however.

One of Auer's observations seems especially relevant to the posted question:

These selected comments [from reviewers at goodreads.com] indicate that let's don't is considered to be bad syntax and a grammatical error that readers do not expect to be used by highly educated characters. The reviewer in [comment] 2b points out that one 'should say "Let's not do that"' rather than 'Let's don't do that'. In fact, it appears that in Present-Day English (PDE) let's don't is not the only alternative to let's not, as we also increasingly come across constructions with don't let's. All three forms can be found both in British English (BrE) and American English (AmE), albeit in different frequency rates. For PDE, Algeo (2006: 30) observes that 'don't let's is 7 times more frequent in British than in American, and let's don't is 4.5 times more frequent than don't let's in American'.

Also relevant to the question is Figure 3.1 in Auer's article (page 50 of Hundt's collection), which indicates that, whatever the relative popularity of let's not, let's don't, and don't let's may have been in the nineteenth century, the print occurrence of all three were dwarfed by instances of let us not—at least in the United States (the chart in Figure 3.1 draws on data from the Corpus of Historical American English).

The decline of let us not (which is roughly one-sixth as frequent as let's not in COHA results from the first decade of the twenty-first century, according to Auer's Figure 3.1 chart) is evident across the nineteenth century and continued across the twentieth century, aside from a significant uptick in usage during the 1940s. Despite this trajectory, don't let's didn't surpass it until about 1920.

By the early 2000s, let us not was not much more common than let's don't and don't let's in COHA results. Still, omitting let us not from consideration can lead to a very skewed impression of what wording was most common during the 1800s and early 1900s.

Least common of all is do let's don't, which appears chiefly (and perhaps exclusively) in comic or parodic settings. From Harry De Puy, "Bull in the Night," in Hemingway Review (1981), cited in James McKelly, "For Whom the Bull Flows: Hemingway in Parody," in Louis Budd & Edwin Cady, eds., On Humor: The Best from 'American Literature' (1992), who cites this an example of parodic language mocking Hemingway's "'Anglicization' of England's English":

"Oh, give a chap a break."

"I say, don't, what?

"What?"

"What?"

"What?"

"I said, I say, don't, what? What?"

"Will I not!"

I say, do don't."

"What?"

"I say, do let's don't. Do. Don't."

"Oh, rather."