I'm not sure why you'd consider this to be a problem. The progress you make is personal, in the form of getting better, rather than being stored on a server somewhere. You don't need additional power rewards on top of that.
I'm curious though, how would you add persistent player progression to a multiplayer game without it massively unbalancing the playing field? The best I can come up with is weapons that are more specialised sidegrades to your characters main weapon, however those aren't also without their flaws. They give you a huge advantage in the niche they perform well at to offset their weaknesses outside of it. If you can switch weapons around between deaths, then you've also just outright given more experienced players more power too.