How does a poker solver select bet size? Doesn't this depend on posteriors on the opponent's 'policy' + hand estimation?

GTO (“game theory optimal”) poker solvers are based around a decision tree with pre-set bet sizes (eg: check, bet small, bet large, all in), which are adjusted/optimized for stack depth and position. This simplifies the problem space: including arbitrary bet sizes would make the tree vastly larger and increase computational cost exponentially.

No, I'm not super certain, but I believe most solvers are trained to be game theory optimal (GTO), which means they assume every other player is also playing GTO. This means there is no strategy which beats them in the long run, but they may not be playing the absolute best strategy.

Nash equilibrium. Optimal strategy for online poker has been known for like literally 20 years right now

Typically when you run a simulation on a hand, you give it some bet size options.

To limit the scope of what it has to simulate.

It's unlikely they're perfect, but there's very small differences in EV betting 100% vs 101.6% or whatever.