The first attestation of the phrase 'fat chance' I found is somewhat problematic, being in the 1778 (second) edition of Helenore; or, The fortunate shepherdess: a poem in the broad Scotch dialect, by Alexander Ross:
Fat chance he furder had, she cud na tell,
But was right fain, that she wan aff hersell.
Some gloss: "fain" means 'glad'; "wan aff" means 'got away'; "hersell" means 'herself'. "Fat chance he furder had" here appears to mean 'what amount of opportunity remained for him', with a slant toward the optimistic 'what large amount'; equally, however, the slant could be toward the pessimistic and ironic 'what small amount'. I'm entertaining objections to the interpretation, either way.
To be sure that the phrase was not a result of typographical error or unfortunate editorial intervention, I checked two later editions of the same work. By chance, I also found that the couplet appears in the Supplement to the Etymological Dictionary of the Scottish Language, v 2, 1825, as attestation of the "wan aff" gloss given in the foregoing (see entry for "To win aff, or off").
The next appearance of the phrase was a nonironic use in Chains and freedom: or, The life and adventures of Peter Wheeler, a colored man yet living, published in 1839:
You see I felt very independent jist now, for I begun to feel my oats a leetle; and so he agreed to give me twenty shillin's if I would, and so I agree tu, and went aboard, and glad enough tu of sich a fat chance of gittin' along.
Couched in ersatz dialect as this is, and supposing the author to be playing a long game, the phrase could be interpreted as an example of the character's misuse of a phrase that would ordinarily be ironic. I think the chance of such authorial subtlety manifesting in this particular work is...not good.
The present sense of the phrase ("a fat lot: a large amount, a great deal: always ironical and implying ‘very little, hardly anything’. Similarly a fat chance, implying ‘hardly any opportunity’", OED) appears to have arisen alongside nonironic or at least borderline nonironic uses, if we take these two, and so rather scanty, attestations as representative of early figurative (supposing the Scottish dialect is not meant to be literal) use, and add into the account later attestations that appear to be nonironic, for example,
Stephen will, I believe do much better in educating the public mind, in using his great talents to manufacture Men, rather than in contriving ways for half men to get at the ballot box. For if he gets them astride his horse, they will jump off and mount the first jackass that bellows out that he has a fat chance at the "crib," and holds out the idea that they may have a chance at the office of currying and feeding him.
Oh! we want men, and men will find a way to act. Haters of slavery, will fight slavery always, but office seekers will fight for office first and not fight slavery at all until it is popular to do so.
Anti-Slavery Bugle (Lisbon, Ohio), 27 Feb 1858 (paywalled).
To be plain, and because there seems to be controversy based on speculation rather than evidence, while it is possible that an idiom with an entirely ironic origin might become established, it is unlikely at best.
In this case: I collected 47 attestations (48 if the suggestive but outlying 1778 Scottish dialect example is counted), from 1839 through 1905, the year before the earliest attestation given by OED (1906) for the "always ironical" sense. The corpora I used were the HathiTrust Digital Library (US and UK), British Newspaper Archive (paywalled; majority of collection is from 1800-1900), and Newspapers+ Publisher Extra (US, aka newspapers.com, paywalled), chosen for their comparative comprehensiveness and relative ease of use. My results can be checked and reproduced using those corpora, allowing for error (horrible, but possible).
Obviously, it is not desirable to display that quantity of examples here, so a detailed timeline will have to suffice.
Year Ironic Not Ironic US UK Total
1839 0 1 1 0 1
1858 0 1 1 0 1
1860 0 1 1 0 1
1867 0 1 1 0 1
1870 0 1 0 1 1
1873 0 1 1 0 1
1886 0 1 1 0 1
1888 1 0 0 1 1
1890 1 uncert 0 1 0 1
1892 3 0 2 1 3
1893 1 0 1 0 1
1895 0 1 1 0 1
1896 0 2 2 0 2
1897 1 uncert 1 2 0 2
1899 1 1 1 1 2 (not ironic, UK; ironic, US)
1900 0 2 1 1 2
1901 2 2 3 0 3 (headline pun = 4: "[Edward] Fat's Chances Good")
1902 1 uncert 2 3 0 3
1903 4 0 4 0 4
1904 3 4 7 0 7
1905 1 6 7 0 7
--------------------------------------------
Total 19 28
What is immediately apparent is that ironic uses of 'fat chance' did not appear in the three corpora until 1888, nearly 50 years after the first nonironic use. Ironic uses were the only uses 1888-1893, followed by only nonironic uses 1895-1896, and possibly 1897. After that, ironic and nonironic uses both appeared in the same years, with the exception of 1900 (no ironic uses) and 1903 (no nonironic uses). It is also notable that, OED declaration notwithstanding, the preponderance of uses in these corpora during the target date range were not ironic.
