As the villain I think betting AA OTT is very thin and probably bad. I feel like most villains that are IP would only go for 2 streets of value with AA, expecially after the top card pairs OTT. i'd expect most villains to check back the turn and value bet AA for value OTR if checked too AND hero call a lot OTR. Betting $250 with AA on a 5
T
Q
Q
($485) can get value from Kk, KJs, (10 combos). villain would have to bet/fold AA OTT, just a way better play to check OTT with AA.
So if villain is competent than I would exclude AA from his range after he bets $250 OTT. Villain is more likely polarized between air or nuts AQ/TT.
Player 1: 40.91% 40.91% 0% [KsKh]
Player 2: 59.09% 59.09% 0% {QQ, TT, AQs, A4s}
If you like the above range than i think calling OTT is fine and unfortunately check/folding OTR, unless you think villain is capable of triple barrel bluffs for 220bb. hero should be calling $250 OTT with AA-QQ, AQ-KQ, and should be raising with TT. Of course, KK is the exact bottom of our calling range so i understand folding it. But if villain either has nuts or air, then KK feels a lot like KQ when villain shoves OTR.