Top 2 in 3 bet pot on straight board

Even given the small sample size, what are your post flop reads on BTN? You only mentioned a few pre-flop tendecies.

Does he bluff, does he overvalue draws or certain made hands? If he is raising with an actual hand, you are probably behind. Bottom two never raises here, that would be insane. He either has the straight, or some FD + SD or a set. On rare occassions he might have just FD and some inside SD or straight blockers. If he has FD and anything else going on, you are flipping. You are almost dead against a set and not getting enough odds to call or GII against a straight.

I think the problem with your hand in particular is the lack of backdoor possibilities. Yes, an 8 gives you the second straight, but other than that you have no backdoor flush draws, no back door straight draws. You are just hoping for a T or a 7.

If you bet pot, indeed some hands with equity might fold, but you probably achieve the same result with 1/2 pot bet. And then you can easily fold to a raise, and you lost less money post. Checking and letting him take the lead is also an option, but here you might need to understand the player more. I personally would prefer 1/2 pot bet and fold to a raise.

I like a 1/3 to 1/2 pot bet. As played, folding.

c/r seems good. As played obvious getin.

It's spot where our range is supposed to check most of time as our high card heavy 3bet range rarely connects very strongly. Especially when it's T76 with two lower cards. E.g. on T96 we can still easily have wraps/oesd with other components.

Now BU on other hand is opening ton of stuff and not folding that much to 3bet. He has way more strong hands and can aggressively stab this flop. Especially if SB is over cbetting and thus usually having even more capped checking range. As you mentioned when you check you often tend to face stabs. That is very effective strategy in general for in position. So what solver does with x-r is giving that possibility for stab from not so strong stuff and then fast playing fairly strong (for low SPR spot) combo but one that has bad visibility on board changing runouts. Some of the stronger combos might want to just x-c as well just to strengthen that part of our range.

Anyway I dislike big cbet the most because you kind of force opponent to play quite optimally. But on other hand this kind of player can still overplay lot of different stuff. In general I would not be cbetting with idea of folding unless I do it because of passive opponent who I don't assume to stab versus check that much and be very honest versus bet. You only need 35% equity after this bet which is not that much. But as said issue is that now you block all those potential stabs with cbet.

I like checking range here. As played obviously call off.

Not folding - we block sets, sucks when he has 89 but he also has a bunch of pair and flush draw combos.