What would you guys do here?

any stats on villain?
generally this is a fold. ur stack is fine afterwards and he doesnt do that with anything u are beating. maybe AK but even if: no reason to flip considering icm.

No stats - he had been at the other table when there were 2 tables. I don't think I've played with him before. Seems to have been pretty quiet so far, not got out of line but we are only a few hands in to the FT.

Given the stack sizes / ICM - would it have been better to just flat the initial raise with JJ and see a flop?

With so many much shorter stacks and given that youre on the BB, flat might seem as a good option. You can try stackingoff on some lower flops and mainly you will have more space to play postflop, so youll get more info about his strenght during later streets.

3bet is also fine, but Id like to mention two things: your 3bet sizing is way too big, Id go for something between 1500-1900. Another problem is that with this 3bet, you are getting yourself in such problems, where he 4bets you and as you see, you dont have almost any fold equity. If you went for a 3bet around 1700, he may 4bet you for something above 4000 and if you 5bet push, you still have much fold equity, plus you make that decision hard for him instead of you.

As played, Id fold, because of almost no info on villain and you say, that he seems to be ABC player. There is no need to flip JJ for 50BBs, mainly on the bubble being one of the bigstacks. Just shy fold, use your edge later (i assume you have some in 25c tourneys :-P ) and try exploiting the bubble and later stages against the lower stacks.

In theory yes, my bankroll is drifting up but I also have a lot of leaks as I am not that experienced. For example in this hand I shoved and got called by AA and bubbled.

Your $Ev in respect to the majority of the stacks at the table is extremely high, flatting would probably be an approachable option, though the optimal play here would be a fold. With your stack size you will be able to exploit less marginal spots later on, given your edge is high in respect to the other players at the table this would also warrant a fold in this situation so that you can continue to push your edge further as the big stack

Do you really think its good to fold JJ against a std minraise when we are 50BB effective? I dont get it...

i don´t get how u confuse a 4bet with a std minraise either tbh.