Look around the chassis carefully, because they often attach one to the metal floor area or engine base.
Does that regulator have the typical 3 wires and then 2 more in the middle of the epoxy? I know some Kawasaki and Kohler engine had that style. On the B&S, the charge light circuit is likely run through a control module with a sense circuit for the charging.
Sometimes, it could just be that the base/low idle is set too low and needs to be adjusted a 1/4 turn so the regulator reads the output of the stator better.
EDIT: I have also had times where I would run a dedicated ground wire to a regulator mounting screw to ensure a proper ground to it.
In any charging circuit the ground is just as important as the positive side but often overlooked.
