Just as the phrase is "for God's sake," the proper grammar is "for fuck's sake." It's possessive. The fuck is the subject here, it's for their sake. For the sake of the fuck.
Technically, the Japanese word "sake" can refer to pretty much any alcohol, although its use in English refers exclusively to the alcohol called nihonshu in Japan.
Also, while you're correct that it is also colloquially referred to as "Japanese rice wine," the drink is actually brewed in a manner closer to beer than wine.
Lastly, while Fuck's sake is great, I'm sure, the best is Bink's sake. Yohohoho
Easier to just use the Japanese word for a thing to mean a Japanese version of that thing rather than adding "Japanese" at the beginning of everything.