Pre flop the 4 bet was a little light, and priced all of the button raising range in, he is paying 275 to win 725, 2.5 return for each coin he puts in, so calling your raise from the button with Any A10+ A suited or connected suited has value. A raise to 525 is more of a standard raise which would be 3x the button bet.
If we are calling to a 4 bet. I think we should be prepared to call the flop expecting to be bet at on the flop. The Flop is certainly in the range of button hitting something, However with the 4 bet pre, I would be putting the button on an over pair to the board. Calling here tho to the flop bet would be a mistake , and a fold or raise to put the button all in is the only plays there. Calling there would do nothing, as would expect the button to barrel again with the last of their chips.

Should we of been in the hand Yes, A K suited on the BB is too strong a hand to fold,
Should we of raised the button. Yes absolutely . The Buttons range is very wide and can often only be a dream.
Should we have called the 4 bet. Here I am not so sure,. with not having knowledge of the player. A 4 bet does narrow his range quite a bit, and it is unlikely that they 9 10 suited or low pairs, and the 4 bet not been an all in shove, I doubt the pocket pair he would be holding are lower then 10's or jacks, You hold blockers to the AA and KK, tho they can still be there.
So the call of 4 bet to see the flop is debatable and more knowledge on the player is needed,, maybe with time you can learn that one, But is not a bad decision I dont thin, we hold strong suited cards, and a flop might be worth to call we are getting priced in at 2-1 with our strong cards.

The flop is a fold or push over him, and let the cards play out ,, essentially hoping for a coin toss from the flop. Good fold I think, only a good call on the flop if you hit your A's or K's and he has Q's