> ... rotating, accelerating ... there is no reason to prefer even intertial frames over nonintertial ones.)
There's a whole set of excellent reasons to prefer to work in inertial frames where possible, most of them having to do with calculational burden. Our case here is essentially flat electrovac with a test field - do you really want to work in Rindler or Born coordinates or worse if you don't have to? You're free to do so if you're feeling masochistic (or doing it as a learning exercise), just as you're free go further and solve Maxwell's equations in curved spacetime in arbitrary coordinates with the vector potentials in some arbitrary gauge (\partial_{\mu} A^{\nu} \neq 0). And with "proton?" then you might as well dive into QCD as well. Unnecessary extra work is surely a good reason to prefer to use the simpler maths where the results will be indistinguishable.
> Without the laws of motion
I don't understand this point. Are you ditching the action principle somehow?
There are no meaningful potentials in the one-particle universe that the OP described. However the particle is "really moving," you could just invent a coordinate system to make it "move" however you like.
Even introducing EM would bump you up to at least two particles! The only reasonable one particle universe would be one where the partice was coupled to nothing.
(It is under the "try to interpret it as they meant it" doctrine that I go from a proton to a particle that doesn't leave behind a disturbed field when it accelerates.)
There's a whole set of excellent reasons to prefer to work in inertial frames where possible, most of them having to do with calculational burden. Our case here is essentially flat electrovac with a test field - do you really want to work in Rindler or Born coordinates or worse if you don't have to? You're free to do so if you're feeling masochistic (or doing it as a learning exercise), just as you're free go further and solve Maxwell's equations in curved spacetime in arbitrary coordinates with the vector potentials in some arbitrary gauge (\partial_{\mu} A^{\nu} \neq 0). And with "proton?" then you might as well dive into QCD as well. Unnecessary extra work is surely a good reason to prefer to use the simpler maths where the results will be indistinguishable.
> Without the laws of motion
I don't understand this point. Are you ditching the action principle somehow?