Both OP's versions are awkwardly expressed, and #2 is also syntactically invalid because there's a missing second instance of the word is as highlighted below...
2a: How many people are there whose reason for [them] knowing who Joaquin Pheonix is is not the Joker?
Also note that although the "reflexive" pronoun them above is syntactically valid, it's not required, and most native speakers wouldn't include it in such constructions.
I'm not sure whether the use of that in example #1 is actually "ungrammatical", but idiomatically it's not good. A better way of introducing the "restrictive" clause (identifying the type of people to be summed up as the "how many" total) would be...
1a: How many people are there for whom the Joker is not the reason [that] they know who Joaquin Pheonix is?
In practice though, the exact question being asked involves a fairly complex restriction, so in a casual conversational context, even many native speakers would struggle to find a concise way of expressing that restriction. It's a bit different for me here because I'm writing (so I have plenty of time to think of how to phrase things), but in a real-world spoken context I think I'd be more likely to ask something like...
3: How many people know who Joaquin Pheonix is apart from because of the Joker?