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Maxwell's equations are a classical field theory (no quantization). That means Maxwell's equations are one of the theories of the body called classical mechanics. So if you wanted to, you could write down the Lagrangian (density) or Hamiltonian for various experimental configurations (eg charged particle in a field) and derive Maxwell's equations (there are a couple of papers like this). Nothing stopping you from using SICM's formalism either. Would it be a useful exercise? No clue.

Sussman and Wisdom do it themselves (briefly) in chapter 10 of Functional Differential Geometry.

They also have a book on field theory, which is just about E&M basically.

Are you sure? I don't see such a book at https://mitpress.mit.edu/author/gerald-jay-sussman-2078/. I'd be very interested to read it if there is one.

Functional Differential Geometry is about the maths required for field theories but focuses on relativity as the main example.

You're right: I'm thinking of the theoretical minimum books by Leonard Susskind and Art Friedman.

https://fab.cba.mit.edu/classes/862.22/index.html

Ch. 6?