AA 3bet pot, bad river card: small bet or check fold?

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AA 3bet pot, bad river card: small bet or check fold?

Hi everyone, have a dilemma about the river
PF standard 3bet
F: Cbet could be 1/3 of the pot, but also could be played as a check (not that big EV deviation imo), but without Ac, I think 1/2 is fine too
T: Good blank card, clear value bet, nothing to say. Ready to shove river
R: Hmm, the river card is too bad, his KQ now turned into straight, TT into set,JT into twopair.. Have approximately 54% vs his range, if he is calling QQ,QJs,KJs preflop. But those hands he would check back most of the time. He has really few bluffs like AcQc,AcKc, some AK, but not that much and even though, I do not have enough confidence, that he is turning into a bluff all of them, so check/call is the worst option imo
Shoving is an overplay, because he is not calling Jx hands or QQ with some super high frequency, So then our equity is less than 50%
Well, we have two options: small thin value bet or check/fold. The small bet, should be really small, I would choose sth like 20% of the pot or so to get call from Jx and QQ ( or actually from KK, AQ and AK sometimes). It may works as a good block bet too, he can just call some twopairs.
What would you do? Can we actually ever check/call river? (I think not
Btw I would more prefer to have KK in this spot (blocking KQ and not blocking AQ)

V: Is unknown, have not full stack, but is playing more tables and seems reggy so far, so I would say we are dealing with a weak-standard reg.

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