I think the turn is pretty interesting, tbh. Probably check-folding vs this sort of player is best, but I could easily see starting to fire here since we got some outs, can rep a straight, and have some nasty rivers to barrel (like the one that actually hit).
But check-calling is bad. Since you put a bet in anyways, I think you must put it in first to get some folds. Tought to get a fold by check-calling
I completely agree with everything you said here. If I'm floating the flop and I pick up more outs on the turn, folding or taking the initiative on the turn are probably my best options. I don't normally take a passive line on two streets vs this kind of player.
PS. Have you ever seen a player call a large river bomb this lightly?