set on draw heavy board

BB in this hand is pretty Laggy preflop with 50/20/2 type stats but seems to play decently post-flop, aggressive, but and doesn’t go nuts with nothing. Button is a good Tag as is MP+1

Hero in MP with 3d3h . 2 limpers, hero limps, MP+1 raises, button cold calls, both blinds call, everybody else calls (14SB’s)

Flop: 2s,3s,4c. SB checks, BB bets, limpers call, hero raises, M+ 1 folds, button CC’s, SB folds, BB 3 bets, limpers fold, hero calls (planning to raise a non dangerous turn) (25SB’s)

Turn: 8h. BB bets, hero raises, button cc’s, BB 3 bets, hero calls, button calls (22 BB’s)

River: 10c BB bets, hero calls, button folds

What I was hoping for here was that he was playing a flush/straight draw aggressively and rather than capping the flop, wait for the turn to raise and hopefully fold out button who might be holding an ace and drawing to the straight.
Maybe this plan is too expensive on this dangerous board. How about abandoning protection, and just calling down after the flop 3 bet?