Sounds like a stator/AC generator fault to me. The stator lives in behind the flywheel.
If a winding is damaged/burnt out, or any of the connections damaged, then the more load you put on the system, i.e. lights, then the less charge that will reach the battery.
Try riding with the lights off and then hitting the "pass" switch. Sometimes this will cause the motor to misfire slightly, or briefly drop a cylinder.
Best thing to do is to pop the flywheel off (use the correct puller, not a hammer!) and remove the stator. Carefully check the windings - maybe even take it to a good local MX dealer to get it checked out.
Andy.
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