[VetteVet]

Your side marker light is a good idea to by pass the light in the cluster or to have a charge light in a gauge cluster but I don't see how it comes on unless the brown wire has a path to ground. You statement that the brown wire is coming out of the regulator seems to me that it is open from there and goes nowhere and it should go to terminal "F" on the alternator or to the no.1 terminal on the internally regulated alternator. On the external regulator the brown wire becomes a blue wire when it leaves the regulator and the sensing wire is white and connects to the "R" terminal on the alternator.

You must have a path to ground in the brown wire if the light is on when you turn on the key and if the light goes out then the path goes away after the alternator starts charging. If the regulator is bad then that might explain it.