Cambridge publishes an extremely detailed description of its math courses (https://www.maths.cam.ac.uk/undergrad/course) which should give you some idea of the starting point. First term courses are:
- Vectors and Matrices
- Differential Equations
- Groups
- Numbers and Sets
And second term courses:
- Analysis I
- Probability
- Vector Calculus
- Dynamics and Relativity
So some of the senior year courses you mention above would overlap, but that would probably be helpful given the intense workload and rigorous treatment. Some of the assumed A level math background (e.g. mechanics) comes under the Physics AP courses in the US.
I think the most useful background would be some of the proof and problem solving approaches typical of AOPS, which I don’t really see in my kids’ US high school curriculum (or even in first year college courses, I know my D got an A in Honors Calc 3 just by memorizing the formulae). So I would seriously look at whether any of the AOPS courses might be of interest. Number theory is an obvious one to consider.
That focus on proofs and problem solving is what we spent our time doing in high school, without needing to go way ahead into linear algebra and beyond. And that’s why AIME, AMO etc are valued because problem solving ability is critical to success there. The exam marking schedule https://www.maths.cam.ac.uk/undergrad/files/schedules.pdf indicates just how sensitive your mark is to the number of “alphas” (which means a near complete solution) - 10 total questions correct across four papers is roughly the borderline between a first and a 2:1.