That flop doesn't suck. Especially against 3bet monkey.

It's easy all-in, because you have only half-pot behind, half stack comitted preflop and possible AK, Ax and other drawing crap against you. Folding that hand in that pot would be absurd.

That's to big of a pot to risk check/check with vulnerable hand. Don't trap here. Go ahead and push, there's no other choice.