What we really need is constitutional carry. This would eliminate the P4P dilemma altogether. While, I OC 99.9% of the time, I like having the ability to conceal. Right now, however, a CHP is the only legal way to do that so I have one.
In regards to funding for the state police, there are a number of viable options. The GA could pass a bill that says all of the proceeds from background checks must go towards completing future background checks. They could auction off confiscated guns. They could staff the background check center with trained volunteers.
I am sure anyone of us could look at the State Police budget and find the necessary funds.