It is impossible to say whether you will be ready to teach multivariable calculus after taking this course. Certainly many of the topics you list will employ some multivariable calculus, but using bits and pieces of an undergrad course as a tool in a higher level course will probably not be the best preparation for teaching.
If you are going to be teaching multivariable calculus, I would recommend getting a few really good textbooks and making sure you can do all of the exercises, motivate all of the definitions and theorems, and feel comfortable enough with the proofs that you can both present the main ideas without any prep work and present a detailed proof with a little bit of prep work. You should also work on having some kind of overarching narrative for the course which makes sense to you.
The textbooks I would recommend are "Multivariable Mathematics" by Shifrin and "Vector Calculus" by Marsden and Tromba.
If you want a "higher level perspective" on multivariable calculus then you should learn some differential geometry. I would recommend John M. Lee's introductions to topologialtopological manifolds, smooth manifolds, and riemannian manifolds in that order.
I want to say that this answer is not meant to discourage you from taking the course you mention. It sounds interesting, and it is often true that we really learn something when we are applying it in different situations. I am sure that taking the course will strengthen your understanding of multivariable calculus, just probably not in the most systematic way.
