Here are my thoughts, guesses, and the small amount of evidence that I have gathered.
The pronunciation of experiment with the "merry" vowel (which is the same as the "square" vowel for speakers with the merry-Mary merger, and the same as the "nurse" vowel for speakers with the "ferry–furry merger") seems likely to be more widespread: as indicated in the original question, it's the main pronunciation given by dictionaries. It's also the pronunciation that would be "expected" based on theoretical considerations: a single vowel letter (other than <u>) in a stressed syllable followed by an unstressed, non-word-final syllable tends to correspond to a "short" vowel in pronunciation (this is sometimes called "Luick's Law"; it's also related to the phenomenon called "trisyllabic laxing", although that name is usually used to refer to some kind of process supposed to specifically affect the pronunciation of certain kinds of derived words, and it seems a bit questionable to me to classify "experiment" as "derived" because even though we can identify an ending -ment and an initial element experi- that occur in other words, both would be bound morphemes; no word like *experi occurs in English as a free base).
I suspect the pronunciation of experiment with the "near" vowel arose either due to influence of the spelling (the pronunciation of "e" in contexts like this tends to be rather unpredictable), the influence of the pronunciation of the related word experience (where the "near" vowel is regular because of the occurence of unstressed "i" before another vowel in the next syllable), or some combination of both.
The phonetic similarity of the vowels might also have contributed to the development and maintenance of the variation. (There are a number of other words of Greek or Latin origin spelled with "erV" (where "V" is any vowel letter) that show variation between these two vowels, such as feral, sclera, query, inherent, coherent, adherent, hysteria—although in these words the "near" vowel is actually preferred by prescriptivists because the vowel occurs in a stressed penultimate syllable, or before unstressed "i" followed by another vowel—and (atmo)spheric(al), for which most prescriptive sources seem to prefer the short vowel, but the long vowel of "near" seems to be common, probably in large part because of influence from the related noun (atmo)sphere.)
I think spelling pronunciations and analogical changes tend to have less clearly defined regional distributions than regular sound changes/mergers, so I am not sure if it would be possible to determine any geographical trends. Like you, I haven't found any source that addresses this question.
The use of the "near" vowel is denounced by Charles Harrington Elster, author of the prescriptive pronunciation guide The Big Book of Beastly Mispronunciations. The relevant entry is available as part of the examples on PBS's website at "What Speech Do We Like Best?":
Experiment ek-SPER-uh-mi̲nt or ek-SPAIR-uh-mi̲nt. The first syllable is often, and acceptably, lightened to ik-.
Do not say ek-SPEER-uh-mi̲nt. Properly, there is no spear in experiment.
Examples from the past
The variant with the "near" vowel doesn't seem to be particularly recent. It was warned against in an issue of the San Bernardino Daily Sun from Monday, August 4, 1947:
There is no "spear" in experiment. The "e" in the second syllable should have the short (eh) sound as in error, errand. Be sure to say: ek-SPEHR-i-m'nt.
[...] Two other words which [sic] the second syllable "e" is heard erroneously as "ee" are severity, as "suh-VEER-i-tee," and sincerity as "sin-SEER-i-tee." But, like the second syllable "e" of experiment the "e's" should be short (eh) as: suh-VEHR-i-tee, sin-SEHR-i-tee.
("Take my word for it", by Frank Colby)
There seems to be some evidence of this pronunciation from the 19th century in the form of the spelling "expeeriment" that seems to have been used intentionally by some authors in dialogue to indicate the pronunciation with the "near" vowel. The context suggests that the authors viewed it as a pronunciation that might be heard from "substandard" speakers.
Here are some examples I found using Google Books, going from most to least recent:
"Now, let's try an expeeriment!" said he, quite in the tone of a Franklin, or a scientific philosopher of modern days. "There's nawthin' like expeerimental conclusions. Jes' you turn your back toward the door, an' I'll turn the lock. There! Did you hear it, sir? Good! hardly noticed it at all, you say? Ah! I thought so; that'll do finely! Well, we'll try the hinges, now. How does it go, sir? Couldn't hear 'em at all, hey? Ha, ha! My expeeriment's bean a suckcess. ..."
(p. 511, The Latimers: a tale of the western insurrection of 1794, by the American author Henry Christopher McCook)
"I guess Thorne's well-meanin' enough," said the other pacifically. "He's a scientific feller, and he's jist wantin' to expeeriment a bit."
"Well, he kin expeeriment all he wants to with his iron and stuff, but I'd advise him to let flesh and blood alone."
(p. 298, Alan Thorne, by Martha Livingston Moodey)
Interestingly, some of the even earlier results for "expeeriment" on Google Books suggest that at one point, some authors associated this pronunciation with dialectal Scottish English, although I don't know if this was an accurate perception or if it was solely based on stereotypes or misconceptions about how Scottish speakers pronounced vowels. (And even if this pronunciation was in fact commonly heard from Scottish speakers in the past, I don't know if that's at all related to the use of the pronunciation with the "near" vowel by some present-day American English speakers).
I found a source ("Overt and Covert Scots Features in Ulster Speech", by G. Brendan Adams) that says that Scots may have the "feel" vowel in some contexts where English has the "pet" vowel, giving the examples "heid" = "head", "sweit" = "sweat", "frein" = "friend", "deid" = "dead" and "weel" = "well". But the words head, sweat, friend, dead all originally had long vowels, and well seems to have maybe had a long vowel in Old English, so the use of [i] in these words in Scots actually seems like a retention of vowels that English shortened rather than the result of any kind of Scottish [ɛ] > [i] change that didn't occur in English. I know very little about this however so this is just my rambling thoughts.
I'm far frae discooragin' ye frae tryin' the expeeriment.' An he baggit the ciller,--ha, ha, ha!"
(p. 31, The Adventures of Mick Callighin, M.P.: A Story of Home Rule; And, The De Burghos, by W. R. Ancketill)
This book is set in Ireland, but the speaker here has a Scottish accent for some reason; on a previous page he says "Cawmill they ca' me in Coonty Doon ; we're a' Scoatch in thae pairts" (p. 26).
But a' the whilk time I ganged o'er the Atlantic, just for the sak of expeeriment, an' travel, and sic like--naething ither, tent ye--for I were weel aff at hame, in Embro', an' might ha'e hangit half the ceety
(p. 4, "The Barber's Letter", Ps and Qs)
One thing I wondered was if we could look at rhymes to see evidence of how people pronounced it, but the short-e pronunciation seems to only have one rhyme, in "merriment" (used by the Victorian poet Robert Browning in the poem "The Glove": "Amid the Court's scoffing and merriment,—
/ As if from no pleasing experiment") and the long-e pronunciation only a possible rhyme for American English speakers in the obscure word "diriment" (for speakers with the serious-Sirius merger) or maybe a nonce-word "cheeriment" derived from "cheery".
Examples from the present
A similar question was asked on the GameSpot Forums, in the thread When did it become "Ex-peer-i-ment"? from 2002. But the participants there didn't seem to know of any regional patterns within the US either. The original poster said:
I've been noticing this more and more, lately. I used to think it was a regional thing in the US, although I wasn't really clear what region it was.
In a later post, he says
I've lived in Boulder, Colorado for most of my life but I'm originally from New York. My "accent" would be consistent with the Denver Metropolotan area, which as far as the US goes is one of the more "unaccented" examples of American English (or so I've been told).