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In the film Battleship, the catchphrase Oh brother, somebody gonna kiss the donkey is uttered by the character JPJ Sailor. The website Aloha On My Mind claims it is Hawaiian slang or Pidgin but doesn't provide a translation.
I did, however, find the word donkey listed on the Hawaiian website Write Pidgin in its introduction it says
Diffren peopo write dea Pidgin language diffren way. Dis how us guys write um fo Da Jesus Book. If you like write um some odda way, az okay, no bodda us guys. Gotta chop up da alphabets so da Internet can bring um to you mo quick. Click da one you like check.
Under the letter -D, donkey is listed and an example of how the word is used is the following
mill stone: two big stones fo make wheat flour. Da small kine, two
wahines can turn da top one togedda. Da big kine, need one donkey fo
So, unsurprisingly, donkey in Hawaiian means donkey in English. Which leaves me with two plausible explanations for the catchphrase.
First, in the UK an ass is just another name for donkey. The American slang kiss my ass! (BrEng arse) to a Brit would literally mean "kiss my donkey"; therefore, somebody is gonna kiss the donkey means that person's request or wish is going to be rejected aggressively. Not having seen the film, I suspect the to-be-kissed-buttucks belong to JPJ Sailor's, in other words, he is flatly refusing to surrender or concede to the "enemy's" demands.
Secondly, but least likely explanation, an ass is another word for a fool, an idiot, a stupid person etc. Hence the phrase "gonna kiss the donkey" could mean that somebody will have to kiss the foolish/idiot/imbecile person very soon.