Ok, I remember the discussions about the bike just shutting down on you. That's still a mystery. The popping/backfiring, though, is usually due to a lean fuel mixture - that is, too much air, not enough fuel. It can come from a number of things; for example, the air/fuel mixture screws are factory set at 1 5/8ths turns - that's lean, and the bike will pop on deceleration consistently at that setting. You might try to turn those out (2 to 2 1/4 turns) may help. But you may also have a vacuum leak - check the boots that link the carbs to the airbox, and the boots that link that air filter housings to the airbox. Those are two areas where leaks are likely to occur.
The stator is underneath the leftside engine cover - there are three wires (yellow) coming out of of the stator that hook into your regulator. If you've relocated your regulator, then it's the three wires that go into the end of that. If you haven't, the regulator is positioned underneath the battery box - look for the three wires going in under there. There are two ways to test your charging system - pull those three wires, and test the resistance across the wires - should be between .3 and .5 ohms. You can also just make sure that your system is generating enough electricity by putting a voltmeter on your battery - should be around 12.7v fully charged. Once the bike is warmed up, and with your rpms up around 3500, you should see the voltage jump to 13.5-14.5v (your system (stator + regulator) is now generating extra electricity - some to run the bike, some to store for "tomorrow"). If those readings all check out ok, the backfiring, etc. is not related to the charging system - so the Team will all have to put our heads together and think a little more.
Wish I could buy your bike - most likely a straightforward fix, although I know it's gotta be really, really frustrating trying to figure out what's causing the problems. Alas - I'm still trying to sell my project bike, so there's no extra room in the stables right now. lol