Flop meh draw in 4bet pot

[Deanglow]

Villain is decent regular. He had been 3betting me quite a bit the first 10 minutes or so (now 20 min in) but has calmed down some since. He won a big pot when he flopped 2pr vs my TPTK. He has won 3/4 3bet pots on the flop with aggression and is pretty aggro postflop in general. I'm gaining some momentum before this:

PokerStars No-Limit Hold'em, $4 BB (2 handed) Poker-Stars Converter Tool from FlopTurnRiver.com (Format: FlopTurnRiver)

BB ($610.85)
Hero ($496.95)

Preflop: Hero is Button with 4, 3.
Hero raises to $12, BB raises to $40, Hero raises to $108, BB calls $68.

Flop: ($216) 5, 5, 6 (2 players)
BB checks, Hero checks.

Turn: ($216) T (2 players)
BB bets $112, Hero

[pocketfours]

cbet

[Deanglow]

but how much

[Numbr2intheWorld]

really seems like a good place to stack off on flop. I'm all for checking behind flops and fuckin people up but you have good equity against basically everything and he's going to fold more than 1/3 of the time. bet size doesn't matter much, i'd just make one where it doesn't look like he has any FE.