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On my 1981 528i, whenever the windshield wipers and heater motor are both on (not uncommon here in the PNW), the radio will cut out for a moment whenever RPMs drop (like putting the clutch in), but then will return to normal in about 1 second. It will even do it if I just let off the gas at the top of a hill.

Once when I had the wipers on, plus the blower on high, plus headlights, the radio would cut out with every stroke of the wipers.

The dash lights are unaffected, and the wipers never seem to change speed with changing electrical load.

Radio is older and aftermarket. Any ideas where I should start? Bad ground?

Yep, start with grounds. I would also test your alternator & battery with a multimeter to see what voltage you're getting. Battery could be on the way out, or the brushes in the alternator could be worn.

I had an old corolla that got really bad voltage drops to the point where it wouldn't start. I had an auto-electrician wire in a relay direct from the battery which fixed the issue.

I am down to the point where my indicator light only blinks when I do not press the brake and have no lights whatsoever switched on. On the positive side, I can see that the indicators are blinking because the coolant temperature gauge jumps up and down with the indicator rhythm

And yes, check for loose/dirty grounds (also alternator-engine), dirty fuses. Might consider to measure the supply to the radio, see if it varies. Check the wiring diagram to see if the circuits share any wiring, that may point you in a direction.