The idea that "kiss the donkey" might be related to "kiss my ass" didn't make sense to me. The context is wrong. So I did some hunting. By 1950 the phrase was so common that it was simply assumed what an author was talking about.
Books: Kiss the Donkey THE MAN WHO LIVED BACKWARD (461 pp.) — Malcolm Ross — Farrar, Straus ($3.50)....If Author Ross had been satisfied merely to pin the tail on his pseudo-liberal donkey, his book might have been a very witty one. As it is, he all too often confuses the silly ass with some of the hard-headed heroes of U.S. history, and starts throwing him political kisses. A man can look mighty foolish kissing a donkey. (Source)
So I made the assumption it might have a World War II or Pre-WWII reference. My first clues came from a transcription from the 1937 New York Radio Daily.
Feb. 26, 1937 J. C. will introduce a game
called "Kiss The Donkey."
and
Mar. 3, 1937 Georgie Hale will be honored at the "Kiss the Donkey" party Sunday in Leon & Eddie's.
So, we have two ideas. Sometime in 1936-1937 there was a game called "Kiss the Donkey" and in early 1937 there was a "Kiss the Donkey" party. I decided to chase the "party" angle first.
An OCR from the New York Daily News, March 6, 1937:
They're playing a game called "Kiss the Donkey" at leon and Eddie's Sunday nites.
OK, so it's not a party, it's a party game!
And here my research stalled. "Kiss the donkey" in a game context has been around a long time, yet while I can find numerous references to its use, I can't find anywhere a game called "kiss the donkey" save in one instance in 1984–1986:
The Odd Couple, neil Simon's new version [of his original play] written from the woman's point of view — Felix becomes Florence and Oscar becomes Olive and the infamous poker game is transformed into "Kiss the Donkey," a game with more wild cards than anyone can keep track of. (Source) (other sources of this play indicate every card is a wild card.)
And yet in the many instances (mostly post 1980) I found, the context seemed to be the same: the ignominy of losing a contest. If that context can be brought to bear (and in my opinion it suits the context of the movie Battleship better than "kiss my ass," which I believe is a taunt that means, "you're less than I am"), then the meaning in the movie is similar to "screw the pooch," meaning "you've lost."
I conclude that the meaning of "somebody is gonna kiss the donkey" is "someone is going to lose."
And if I had the 16-inch guns of the U.S.S. Missouri pointed at me, I would be inclined to agree.