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.