it sounds like you may have a coil pair in the stator shorted , or open circuited, not producing full amperage with headlight on.
There are 6 coils in the stator...each pair that are diametrically opposite are wired in series, giving 3 such pairs. one pair is brought out on one of the stator leads ( can't say which until I refer to my manual at the office) and this is for daytime running without lights and produces 1/3 stator output. The other 2 pairs are wired in parallel and are brought out on the other stator lead, giving 2/3 output. The headlight switch selects 1/3, 2/3, or 3/3 output depending on position of switch.
There is no over voltage protection....I put in a switch to open-circuit the stator for daytime running after battery boil over took off chrome on primary case and left muffler. With a magneto, battery is only needed for brake light in daytime, unless you are in a jurisdiction that requires MC's to run lights in daytime. I have changed to 12 volt...only requires changing the lamps and battery....the stator will put put sufficient voltage if all coils are patent. I would use one of the voltage regulators that are now available, rather than a Zener.
If you have a good VOM, test continuity and resistance between both stator leads and common lead (brown one I think). This is not an absolute test, as the parallel wired coils can have a fault that is masked. A rough indication that all is well would be open circuit between all three leads and the MC frame ( no shorts to ground), then resistance between one of the green leads and brown is half (or twice) the other green and brown. The parallel wired pair should have 1/2 the resistance of the single pair. An open circuit would indicate a broken wire in the coils or an open in a lead wire.
A more thorough diagnostic would be to disconnect the three stator leads, then with engine running at say 2000 RPM, measure voltage between each of the green stator leads and the common (brown) lead. The voltages should be the same if all coils are patent. A low voltage value would indicate a new stator. This must be done on AC voltage setting....it is an alternator after all.
Pm me if you need those wire colors coming off the stator.