Wouldn't application of brake lights result in a constant amp meter reading vice oscillation? Were as a turn signal or flasher has the constant spiking of the load as the relay makes and breaks contact.

You could test this with car off. Just on the battery and see what type of current load you get on the brake lights, and the turn signals. If it is high chances are a possible short to ground that is amplifying the current draw. If things are normal on the battery alone, then I would look at alternator and VR. It is possible alternator is not keeping up and your seeing a heavy swing as it flips between alternator sourced and battery sourced. Alternator can be tested at local auto parts.