And now there's an (almost?) impossible-to-remove
stuck-on maybe two-inches-square piece of fried-on
plastic on inside and outside.

QUESTION: short of putting a metal-file to it (and
scratching up the nice finish), HOW TO REMOVE IT?

Scrubbing doesn't work, Nor does trying to (carefully)
use a sharp knife and somehow get it off that way.

Depends on what the pot is made of.
Solvents(Acetone?), fire(Propane torch),
abrasion(fine sandpaper steel wool, or a wire brush),
and cold (Like, dry-ice cold. freeze the spot
until frost starts appearing on the rest of the pot,
and then whack it with a stick.)
are your choices;