There are lots of great CM books out there. Besides Taylor and Gregory some of my favorites are
- Scheck - Mechanics: From Newton's Laws to Deterministic Chaos
-Lanczos - The Variational Principles of Mechanics...
In my opinion the exercises are not that hard. I think there's a good balance between computational exercises and proofs... you souldn't have much trouble figuring out whether your answers are correct or not.
I'm really surprised no one has mentioned Gauge Fields, Knots and Gravity by John Baez...
I love that book, it's not very rigorous, but it provides a lot of insight, and for each topic gives a list of references, so that...
I'm confused haha.. you dislike hand-wavy books, but you are not looking for a pure math book...
You should also note that Hassani wrote two books, Math Methods For Students of Physics and Related Fields (maybe the one you looked at), and the one I mentioned earlier, which is a graduate...
Hassani gives a really nice treatment of Green Functions in his book
Mathematical Physics: A Modern Introduction to its Foundations
You could try
Ordinary Differential Equations by Arnold
or Differential Equations: A Dynamical Systems Approach by Hubbard & West
They're both great!