if you're checking all 3 streets behind then your thought process is "i have SDV and want to get to SD as cheaply as possible". that doesn't mean we can't call a stab or 2, depending on board texture and villain's tendencies. so if you think they try to bluff you off then simply call some of your better hands for getting into that spot. for example here, if you DO play AK this way (i.e. don't bet flop for val+protection) then it is one of your best hands for checking back, so you can happily call his turn stab as you did. if the turn is a Q then it's not so nice here because that hits his range (at least i'd expect an nl2 SB to call here with a range like this {JJ-22, AQs-A7s, KTs+, QTs+, JTs, T9s, 98s, AQo-ATo, KJo+}, maybe tighter maybe looser - we don't know). So i think we like turns that brick (somewhat), like 2, 4, 6, 7, 8, 9 that are not :spade:. with that said in this particular hand it probably makes more sense to bet flop for value and protection - we probably have 6 outs anyway which is ok. Basically, high cards favour SB in this spot since you can see on this flop that ~70% of the range i gave him is simply overcards on this flop. so he improves on a Q. we could call once if we expect him to stab but folding won't be a problem. much better to call on one of those bricks though.
i don't really know what to do if he checks again in this case because looking at it, this board is going to be getting much worse on later streets so it's very likely to bring cards we don't like (T/J/Q/ spades, backdoor FD, turns and rivers that will give him weird straights with his PP etc). so basically if we are playing a SDV line like checking back then we shouldn't be betting. in this hand if i made it to the turn checking and then villain checks i'm going to start hating myself for checking flop and not just bombing it and taking it down there. anyway, if we check back and then he stabs river it really feels like we should fold. he's played a SDV/give up line all the way down to the river and then he bets it looks like he hit something. so yeah i agree he probably would have bluffed earlier.