could definitely overcall, but BTN is the spot of the table and I'm deep with him, so I choose to squeeze with a very playable and powerful hand, as HJ is folding to 3b a decent bit.

IMO vilain is opening 26% or so in this position, and he's folding a lot to 3b in general. I wouldn't expect him to flat here with KQo or AJo. Probably more with all pairs between 77 and JJ (possibly some slowplays with the fish behind, but really unsure about that), suited broadways, and AQ.
For what I've seen he seems rather tight and straightforward, but I have a not saying: 'randomly opens K8o CO and C-bet pure bluff 3 way'.

Overall, considering how I've been playing in the session, I think I'll look rather strong when I squeeze in this spot.

Check-folding here.

I'm doing a lot of delay C-bet in 3b pots when OOP these days, so when he checks back I believe it's decent to take a stab. I'd play AK this way for sure (these days), and also AA and KK.

River seems kinda bad for me in the sense that I should have very few straights and flushes. I would still have hands I'd want to value bet, but I'm capped, whereas vilain has some nut combos. I'm not sure to which extent he'll exploit the situation though.
Should we continue on this runout? 136€ eff behind. IDK if I could have only one sizing here (that is, shove?), or if I should bet half pot, 1/4th pot? Or just have a X/F range and a X/C range.