My perspective on anthropics is somewhat different than many, but I think that in a probability theory textbook, anthropics should only be treated as a special case of assigning probabilities to events generated by causal systems. Which requires some familiarity with causal graphs. It might be worth thinking about organizing material like that into a second book, which can have causality in an early chapter.

I would include Savage's theorem, which is really pretty interesting. A bit more theorem-proving in general, really.

Solomonoff induction is a bit complicated, I'm not sure it's worthwhile to cover at a more than cursory level, but it's definitely an important part of a discussion about what properties we want priors to have.

On that note, a subject of some modern interest is how to make good inferences when we have limited computational resources. This both means explicitly using probability distributions that are easy to calculate with (e.g. gaussian, cauchy, uniform) , and also implicitly using easy distributions by neglecting certain details or applying certain approximations.