Why would you ever even consider flatting 4bets with AK? Do you love to get yourself in tough spot on 70% of the flops for the fun of it? Especially with a hand thats an easiest 5bet ship ever?
If you wanna call 4bets that really depends on his 4betting range and his sizings. In your scenario with those sizings you really can't do much but shove or fold.
The spots where you can have a flatting range and min 5bet range are BU vs CO when CO opens to 2.5x, you make it 7.5x and he 4bets small to like 16-17x which a lot of ppl do.
The second one is when you 3bet a 2x open from the blind to 7.5x and he makes it like 15-16x.
Those 2 spots are really the only common spots where you should be 5bet/folding where you can make it like 25bb and still fold. If you make it 34bb like you suggest you have effectively committed yourself if he shoves and you can no longer fold at that point.
Obviously all of it also depends on their 4bet range %%. Your really looking for at least a 10%4bet range or higher when considering 5bet bluffs and flats.
As for the hands you should flat are, apart from AA and KK if you think flatting those has higher EV, your tier 1 hands should be KQ, AJ and AQ. The reason behind those is, if he is infact 4betting at least 10%, if you flop a pair with those hands then you have about 70-80% equity on the flop against his preflop 4betting range. If the flop comes K76 and you hold KQ then you basically have the nuts. The only hands that beat you at this point are AK,KK and AA which make a total of 2% hand range and there's a bunch of blockers out already. This means if his 4bet range is 10% you have him beat over 80% of the time. Whats even better is he might be using Kx hands as 4bet bluffs and now his confused what to do with TPNK in a 4bet bloated pot.
Same goes for AJ and AQ, you just manage to put him in a really tough spot when he is using all the Axs as 4bet bluffs and you both flop top pair. Its definitely a spot where i find that regs make mistakes and AJ and AQ are even better since the only hand your worried about is AK and AA which you again have blockers to and they usually just flat AQ so AQ and AJ are similar hand strength in this spot.
You can also flat suited connectors if someone is a maniac who 4bets more then 15% in specific spot, your plan being to flop some draw or a pair on a low board like with 56s on 952 flop. Your plan in this spot being to just jam over his cbet as his range is so wide he will almost never have a pair in this spot since overpairs are such a small % of his wide 4betting range that he will be forced to fold a very high % of the time and even if you get called by AA you still have 25% equity, but this is really getting into spewy territory
Not that thats bad because if someone see's you flatting 4bets with 56s and then shipping it once you flop a pair thats really going to help your overall image