Here are my thoughts, guesses, and the (not very substantial) 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. (Some other words can be pronounced with either of these vowels, such as feral, 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)
I also found an example of it apparently being depicted, in a patch of dialogue filled with indicators of supposed mispronunciations, in a book from 1898, The Latimers: a tale of the western insurrection of 1794, by the American author Henry Christopher McCook:
"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)
It also occurs in a similar bit of dialogue from Alan Thorne, by Martha Livingston Moodey (1889):
"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)
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).