I've been using either MPro7 gun cleaner in the spray bottle (they also sell a copper remover) or Hoppes Bench Rest 9. The MPro7 is supposed to do it's thing in 5 minutes and is virtually odourless, the Hoppes needs to sit in the barrel overnight and stinks to high heaven. I don't know if one is better than the other, my wife much prefers the MPro though.
Both seem to produce silky smooth clean barrels.
I don't shoot a lot of center fire jacketed bullets other than 9mm, I do usually see some blue-green residue out of that barrel when cleaned with either product.
The center fire target/precision forum will likely yield better results on your copper remover question than here, though many of us do shoot jacketed bullets too.