Besides for rote memorization, is there some rule that determines when the 人 kanji is pronounced as にん or じん?
Not universally. にん is the reading for counting people above two and じん is the reading when it’s used as a true suffix (such as for nationality, race, etc.). Beyond that it’s mostly about when the word was incorporated from Chinese. Most of any other “rules” you see on this board are ad hoc rules of thumb that are prone to exceptions.
Your best bet is going to just to come up with effective mnemonics for each word.
No there is no rule. Basically the reading depends on when it came into the language from China. You just have to memorize it.
I have started strongly visualizing either a NINja all in black or a dude/dudette with super long hair named JIN for each respective vocab.
So, for example びょうにん / 病人 is a -NIN ending so I imagine how weird is it for a NINja to be dressed all in black when a Sick Person (imagining him all in black in a sick bed/hospital bed) when those are usually associated white sheets and white walls.
Or for きゅうじん / 求人 I imagine long-haired JIN, looking bedraggled with that overlong and unwashed hair, unsuccessfully responding to a Help Wanted sign.
Etc. Good luck!!