you COULD run it.. I'd run a seperate wire circuit for the back. the problem is you will almost assuredly be in a discharge situation with all 4 burning, and running.
for short time period.. ok.. but for extended it will overheat that generator... and that's when run at rated rpm to actually get that wattage / amperage out of it. running at lower throttle you will reduce genny load.. make it happier.. but will also be discharging the battery slowly.
COULD do that. IE.. idle down to 15a draw to keep genny happy for long term.. and be discharging.. then before shutdown, kill rear lamps and throttle up to rated rppm and run around for another 20-30 minutes to recharge storage cell.