CUC is what I would use, and I don't think that I would scrub the embroidery area as much as I would dab with a MF Utility towel.
You may want to try a inconspicuous spot to make sure that the red doesn't bleed to the white.
First off with new paint that's baked on you don't need to wait.
As far as BG first or PS first, it comes down to a personal preference......
I always apply BG under PS....it does not affect the longevity of your PS.
I even take it a step farther I apply BG over my polish residue and wipe them both off with one swipe, I get the benefit of the BG and the polish residue wipes off easier.
That being said if you feel strongly about applying your PS first by all means do it.