Prime Video’s “Ricky Stanicky,” the latest comedy from director Peter Farrelly, is filled with outrageous moments revolving around John Cena as Rod, a down-on-his-luck performer hired by a group of friends to pose as their nonexistent pal. The undeniable highlights, however, are the moments in which we see the Atlantic City lounge act for which Rod dresses as iconic performers, from Peter Frampton and Britney Spears to Billy Idol and Boy George. The musical numbers walk a fine line, presenting Cena with costumes that immediately summon up associations with their sources while remaining just a bit off in order to show that Rod is not exactly one of Atlantic City’s top acts.

Cena credits the costume and hair and makeup departments with creating a wide variety of hilarious looks — and doing it on a ridiculously compressed schedule. “Not only did they transform me to look like Dee Snider,...