So a few things. BU can call wide in position, but let's chill out thinking a lot of players 4-bet JJ and 10's at $1/3 so those hands aren't "likely". So please include those in his range. Next, ask yourself, if you had AA, KK, QQ, JJ or 10's would you check this board? Or does it look like you bricked with AK, even though you didn't really. NEXT, if you did have AA, KK, QQ, JJ or 10's, are you really gonna check raise them?
These are a few good questions. Next, if you giving him a set of 4's, I'm giving him all PP. And if he's calling a 3-bet with all PP, then he's calling with a lot of wide broadways. So let's look at his range.
MP2 61.03% 58.04% 2.99% AdKd
MP3 38.97% 35.99% 2.99% QQ-22, ATs+, KJs+, QJs, JTs, AKo, AJo, KQo
That's pretty good equity, but no made hand, which means.....bet the flop dude. You have FE that knocks out random crappy hands that are beating you, you improve on a ton of cards to come, and now your playing "oh geez what does he have", don't really have many credible C/R's in your range that make sense. So let's give him the worse case scenario, and give him only sets of JJ and 10s.
MP2 32.90% 32.90% 0.00% AdKd
MP3 67.10% 67.10% 0.00% JJ-TT
Your a 1:2 dog, so how do you make up for this? You gotta bet this flop. You need to maximize profit from the other times he has garbage, and will fold, because this hand CAN stand up to a lot of pressure. So say you bet this flop and get raised, what hands will he realistically raise with? AJ? That's dumb, KK and QQ don't fold here, and so if he get's re-popped he's screwed. QQ? Maybe, but QQ doesn't love life if you raise here and you flat, cause uh that's warning bells of AA, KK or JJ-10 (which you also contain you in range). So this guy really shouldn't be raising a ton, but he may call a fair amount. But that's cool, cause we got a lot of great cards to come, and his "calling hands" really can't stand up to a ton of pressure. You calling down 3 barrels with AJ? KJ? QJ? A10? See why this hand is a bet, bet bet hand? Now instead, your stuck holding exactly what it looks like your holding, AK or AQ. You bet the flop, and he smashes a raise in on your, yeah, I'm kinda a bit scared, cause his only real raises here should be strong combo draws, like QJdd, and sets and 2 pairs. But look at all those other hands he has. 43 hands, 11 of which are awful for you, which you'll probably hear from on the flop or the turn (if you don't already have the nuts). So barrel away my friend.