Can 'An ass that won't quit' connote stubbornness?

Solution 1:

I'm not trying to make you the butt of the joke: I agree that in this one case your conclusion is refreshingly naïve. An ass that just won't quit is callipygian, not equine.

I have Juba to Jive: A Dictionary of African-American slang open to won't quit:

outstanding; great; truly beautiful.

It's hard to disprove a negative, but I simply cannot idiomatically read "ass" in your text as relating to stubbornness. (A commenter discovered empirically that every ass that won't quit was a bottom.)

McWhorter's Nine Nasty Words, which I am finishing today, tells how "ass" jumped from donkey to buttocks, at least in America. Also, see there a discussion of "ass" as a pronoun, along the lines of the synecdoche you mentioned.

You've seen several clever rephrasings in the comments but, in the end (sorry), your version doesn't have the effect you're looking for.