Hmmm. not likely. Usually these work by the engine spinning a magnet around within some stationary coils of wire. Simply put, this moving magnetic field is what creates the current to charge the battery. incidentally this same moving magnetic field is how it works in a car alternator too (heck for that matter, the Hoover dam), but thats a different topic.
It's not likely that your fuel delivery system (although something IS definitely wrong with your fuel delivery) is the direct cause of your charging issue. Although running on one cylinder is usually not good for longevity either.
It's like wondering why your radio won't turn on is because you're out of gas.
I know that some older bikes are lossy under a certain rpm range, the early CBs for example. You don't really get any useful current off the charging circuit unless you're over 3k or so, so it may be something similar with yours.
How do you know it's not charging? Is it simply that you keep coming back to a dead battery? Or did you test the current coming off the stator?
My best advice would be to fix the problems as you see them. You know it's only running on 1 cylinder so fix that problem. then when you got it running sharp with both, figure out if you have a charging issue. Test the voltage and current at > 3K. It should at least be within spec at this rpm. otherwise, it may simply be that you can't keep the engine rpm high enough to do any real charging.
Hope some of this helps.
