start with a fully charged battery and measure these voltages
1 motor off/ignition off - voltage across battery terminals
2 motor off/ignition on - voltage across battery terminals
3 motor on and headlight off
measure the voltage at 1000 rpm, 2000rpm, 3000rpm, 4000rpm (hold the revs for 15 seconds at each rpm or get a maximum reading)
4 motor on headlight on
measure the voltage at 1000 rpm, 2000rpm, 3000rpm, 4000rpm (hold the revs for 15 seconds at each rpm or get a maximum reading)
5. motor off/ignition off - voltage across battery terminals
what are the readings

and from what you initially describe it sounds like you have a short in the wiring somewhere..