OED online has a wider second definition than that given in the question:
2. A great or skilful hunter (freq. ironic); any person who likes to hunt. Also fig.
This "frequently ironic" may be the transitional clue between the great hunter of old and the stupid or contemptible person of today, first quoted by the OED in 1933.
The 2008 New Partridge Dictionary of Slang and Unconventional English says:
nimrod noun a fool, a stupid person, a bungler. Jonathan Lighter writes that ‘currency of the term owes much to its appearance in a 1940s Warner Bros. cartoon in which Bugs Bunny refers to the hunter Elmer Fudd as "poor little Nimrod"’. It is not clear that watchers of the cartoon understood the C18 sense of the word as ‘a great hunter’, but the term has stuck US, 1932
The OED's 1933 is somewhat ambiguous, it could be referring to a bad hunter:
1933 B. Hecht & G. Fowler Great Magoo iii. i. 183 He's in love with her. That makes about the tenth. The same old Nimrod. Won't let her alone for a second.
Their next idiot quotation isn't until 1963. However, etymonline.com isn't convinced by Bugs Bunny changing the meaning:
It came to mean "geek, klutz" by 1983 in teenager slang, for unknown reasons. (Amateur theories include its occasional use in "Bugs Bunny" cartoon episodes featuring rabbit-hunting Elmer Fudd as a foil; its possible ironic use, among hunters, for a clumsy member of their fraternity; or a stereotype of deer hunters by the non-hunting population in the U.S.)
As it happens, Nimrod is also given as one amongst two whole-column-lengths of synonyms for penis in Farmer and Henley's 1891 Slang and its analogues past and present.
The 1902 edition defines it:
NIMROD, subs, (colloquial). — I. A
hunting-man ; a sportsman.
- subs, (venery). — The penis.
[Because 'a mighty hunter']. See
CREAMSTICK and PRICK.
Perhaps not relevant, but from the same volume:
NIMENOG, subs. (old). — A fool. Also
NIGMENOG.—B. E. (1696).