[checktheriver]

On such a wet flop raising is mandatory unless you have a specific read. There are a lot of draws in his range, he might be donking / calling a raise with 99-TT or even TP. I'd raise more though (especially given it's multiway even though BTN is folding pretty much all the time unless he has a monster), make it ~20 and call a shove.

As played I'd fold turn, but don't you think he'd just push the flop with a set ?