Up until very recently, I used to make similar folds to you, and in a sense I am very happy to see "someone who plays like me" because it makes me see where it goes wrong when somebody else does it (too).
From my point of view, I always assumed: how can anyone reraise (or in this case flatcall) here without AAxx or it must be AAxx or what else can it be...in my personal experience, it caused my brain to stop working and even though I always prided myself in thinking I can put my opponent on exactly the 2 cards he will be using out of his hand, having been wrong so often I forced myself to just give it up altogether and simply try to think in ranges instead of just one specific hand my opponent would do this with. Of course you can say this but only if you KNOW for sure (you would need to know this opponent very well to know he would only do this only with AAxx) because everytime he would only do this with AAxx. This is obviously not the case here. Trust me when I say I have put down middle sets to see my opponent take it all with top two or 3 pair. I get so upset nowadays from nitfolding the best hand that simplygoing all in with middle set in THESE kind of situations and actually being up against AAxx doesn't phase me at all compared to the feeling I get when putting down KKK here vs top two; I used to feel great when I saw that I was right and my opponent indeed had AAxx but I have been on the wrong side more often so I stopped doing it; of course, AAxx is part of his shoving range, of course he would do it with AAxx as well, but he obv thinks that he is blocking both AAA and KKK with his AKxx so mayybe that is why he is so "fearless" and he thinks ah well if I am up against a set so be it but I am blocking both top sets so maybe we can borrow some of his "so be it" attitude even though from a nitty point of view, going all in with top two is "wrong"; nowadays, I just stop and think first before pressing fold, and unless I know deadcertain he would only do it with AAxx and NEVER with anything less, I would fold, otherwise I just ship it as I believe I am ahead against his entireshoving range and say fuck it when he has it; playing with a lot of buy ins for a certain level has helped me a lot with this. Once again, just because we fear the one hand that beats us the most, it doesn't mean he has to have it.
Also I recognize in your game a huge mistake, that is I used to make it too, you just flatcall with KKK but NOT to trap, no to be able to still bow out when somebody "shows strength" and that is what you did. After investing $1,10 more to the pot you should really say screw it I am not folding anymore nomatter what happens. Here I would just say against a shoving range of AAxx, 33xx, and AKxx and A3xx,, my KKK looks pretty solid : ) add the fact that you added $1,10 into the pot already and that it is a 3 way pot (bb calls before it is on you) I would even consider the BB also to have an Axxx type hand which would if it were true make AAxx super unlikely now, so I would never fold anymore here, but like I said, used to make these folds all the time until they started tilting me beyond belief. For me, the problem was also very often that I hated the idea of the possibility of losing a huge amount of chips (both you and bb have pretty deep stacks and I would have been more inclined if the smallest effective stack had been 90bb or something) but now I just smile at the opportunity to make an even bigger stack. The problem here is also that I would raise here with KKK or call to trap, but not call to fold, but I can imagine your point of view because I think that if you had raised, he would still have reraised you in all likelyness, which would then only make you more certain he has AAxx as now we are talking about his 4betting range which would then be increasingly tight and would severely lean towards AAxx. but like I said, villain thinks top two on this board is the nuts, take a note oh him but be careful with these kind of laydowns as perhaps we both learned in plo you have to draw to the nuts only (that is the beginner advice that I got and also from reading hwang's first book on plo which is too nitty for 6max I think) but if our opponent's dont share that point of view but we keep hanging on to that point of view regardless, we are costing ourselves a lot of money.
Good Luck