Re: dumb polishing/compounding ?
IMO, you don't have to strip your LSP's before you use the UC (this will be heresy to some). Also, although some clay can be abrasive, I'm pretty certain that regular (medium or mild) clay won't remove a measurable amount of clearcoat.
I'd use a microfiber applicator with the UC, maybe a foam with the UP.
"Spray it like you mean it!"