I haven't used any fall out products on my JCW and I haven't used any Carpro products so can't help there, sorry.
When TK Tom detailed my Wife's cooper, he used a Waxtec Fallout on it, but only on the wheels and the back.
If it was me I'd possibly wait till it gets a little warmer to do the full works (wash, clay, polish, wax etc) on them. I did mine a couple of weeks ago (minus the clay) and it took ages as the cold and damp were effecting the wax.
For better advice it maybe worth a quick call to TK Tom and get his opinion