This is the stain I used.
Appears as though it could be fine leaving well enough alone. The shininess is likely due to the newness of that coat, and the fact that I was liberally slathering the stuff on, probably in violation of the manufacturer's directions.
But like I said, I'm going to give it a few days before I sand anything.
Thank you for the advice on the paint. There is definitely vinyl sealer under the white coat, so the body is sealed (I'm also paying attention to the finishing discourse over on the other build diary)
Now the waiting game. I have to learn patience at this step. Any advice on how to pass the next few weeks? Puzzles? Netflix?