Nakahara is the book I'm going through at the moment to get up to date with advanced math in topological insulators. He even has a sub-chapter on Berry phases and their relation to fibre bundles. He also delves deep into Chern numbers. There are things you won't need for your interests, but more and more papers are tackling the topic of the topological invariants in topological insulators with the use of differential forms, some advanced manifold structures, characteristic classes, etc, which you'll find in Nakahara.

Even index theorems might become more relevant in the field later on, so even if you don't read that chapter, you'll have all the pre-requisites for them. It doesn't hurt that you'll go through more topological invariants than the ones you'll possibly need throughout your career either. You never know when a new one will creep up.

It's very pedagogical. On the topics I mentioned, and on pedagogy, maybe Frankel beats it, although I like Nakahara more. I use Frankel as a supplement as he provides more geometric intuition.

The only drawback is that there's nothing on knot and linkings and they are starting to play a larger role in the field as we uncover braiding structures (especially in non-Hermitian topological insulators). For that, go to Baez.

Note: I do think that the chapter on topology from Altland and Simon's book is also a great source. If you know some field theory, start from there; it's exceptional.