"The Anacreontic Song" (or "To Anacreon in Heaven"), with various lyrics, was already popular in the United States

Is there any evidence for this?

It was never very popular in Britain, if the number of reprintings it didn't get is any indication. Artificially twee text, clumsy and nearly unsingable tune - a song destined for oblivion if ever there was one.

Isn't it more likely that it was something that just Key and his dog were fond of?

(It could have been an early instance of a dominant culture using a subordinate one as a cultural colostomy bag. We dumped "To Anacreon" on the Americans, then when it was payback time they dumped Sinatra, Dylan, Stephen Sondheim and Lady Gaga on us).