I actually think your play is fine preflop but if you are worried about flatters then you could shove as your stack is less than 10 times villains raise and just the blinds, his squeeze and the initial raise + call is worth at least 20-25% extra on your stack which is valuable without having to run a race.

Having said that, I actually like your 4 bet over the squeeze as it could induce the guy with the big stack to make a play at the pot preflop hoping you might fold a hand like AJ/AQ or some medium / high strength pairs that you decided to test the water with but might fold to a 5 bet shove seeing as you would still have 25-30bbs left if you did fold.

Once he flats you here you really need to know the guys tendencies. Has he called after 3 betting before? Does he flat here with monsters? I really cannot see him flatting with suited connectors unless he is really bad and he would surely fold hands like KQ/KJ/K10. Pocket pairs from about 88 and below I think fold to your 4 bet as there are no set mining odds and AK shoves all in against you so I think the guy has AJ/AQ type hands, a pocket pair 99-JJ where he may jam if the flop comes low or he hits set but will give up if the flop hits your high card range. Alternatively he could be trapping with a monster AA/KK in which case you are in trouble.

I think as played I would shove the flop after his check. If he has you beaten with AA/KK/AK so be it. If he has 99-JJ he may decide that its good enough to go with anyway given that only one high card beats him and you have huge value if he ever does that. With 35% of your stack in preflop I think you are committed to the pot at this stage and you would be betting 140k into 200k which is by no means a huge bet. I think I would try to win it now and hope he heroes with a lower pair or we suck out if he has us beaten.