The shop installed a ground loop isolator. So far, so good.
But to follow up, the noise wasn't every time I started it, but I'd estimate 2/3 of the time. I never timed it exactly, but was always after the 2nd or 3rd song, so I'd say 7-12 minutes, but never before. I never tried turning on the audio system for 10+ minutes without starting the car, so I can't answer that. Before or after the noise, I tried turning on and off every electrical device and could never force it happen or stop. It's been very hot the last 2 weeks (mid/upper 90's) so fans + A/C surely would have kicked in by then.