Kip,
Balder-Dash, Poppy-Cock, and HarUmph, I say, to the 12-volt comment. The problem isn't that it's a 6-volt system, the problem is that it's a 6-volt system that's in need of repair! Find the problem, fix it, and move on.
Now . . . what was the problem? . . .
Oh yeah, right . . . the "get up and go" gets up and goes after 10 minutes. Hmmm . . . I like the "leave the charger on it while it runs" idea. It's definitely time to get out the volt-meter. See what voltage at the battery is when running, check it at the regulator. Then, how about a "full-Field" test, to check the generator?
My regulator started acting a whole lot better recently after I doused it with contact cleaner from Radio Shack. The shop manual will have all sorts of tests for the regulator and electrical system.
AND, tightening the "coming loose" wire from the ignition to the coil helped heaps, also! Always good to follow the electron path, and check all wires and connections.
Stick with it, it will turn up sooner or later . . .
-Michael