Quote - Does what I'm describing make any sense?  Or is it nonsense?

I may well have misunderstood. Once you've repositioned* the texture for one figure to fit another's mapping, why would you need to render it? Why not just apply it?

I do remember seeing a suggestion by _dodger somewhere that involved morphing a plane which carried the texture for one figure so that it fit the mapping of another, IIRC. Then you would have to render the morphed plane to get your repositioned texture. I don't think anything came of that, though. It sounds easy until you think about what's involved. For all I know, that's how UTC works at a conceptual level.

*I'm using that term instead of writing remapped, as you did, because I hope it causes less confusion. Most people take remapping to mean altering the UV mapping of a mesh so it will fit the texture, as BB and others said.