*Aside: when and where do US students learn the stuff that's in our Mechanics modules - kinematics, rotation of rigid bodies, simple harmonic motion, all that stuff? Physics? University? It doesn't seem to fit into any US school maths course I know about.

I'd expect to see these topics in physics. I haven't yet looked at the AP Physics C curriculum in depth but I would expect these to be covered there, in the mechanics half obviously, rather than the E&M half. Personally, I was very disappointed in the lack of math in DD's middle school science, don't know why I expected it to be different from when I was a kid though. I don't know what the extent of the math in other high school physics classes would be - usually kids have to take calc before or concurrent with AP Phys C.

I think I would lean toward getting a lot of math via applications - really getting into mathematical models in science, for instance. Given the number of years... maybe alternate between years focused on learning math and years focused on learning how to use math?