"make an ass of one's self," or "to be an ass" to me at any rate clearly mean to resemble a donkey. As in Midsummer Night's Dream.

"I see their knavery: this is to make an ass of me...I have had a dream, past the wit of man to say what dream it was: man is but an ass, if he go about to expound this dream."

I suppose it could be that Webster, or whoever, adjusted the spelling to match the way Americans pronounced the word. I'd think it more likely that there was an element of euphemism in it.

In many parts of England, especially the North where the short a is more prevalant, the way people pronounce "ass" and "arse" is completely different. Even in the South I've noticed that people often tend to use the short a when referring to donkeys - maybe to avoid confusion. I take it that in America people will normally use the same pronunciation whichever sense of the word is intended?