[Ultramaroon]

That doesn't sound like clutch/TOB. Too regular and too high of a frequency at idle.

I know it sounds lame but I keep going back to something driven by the accessory belt.

edit: I just went back and reread your OP. Heaviest electrical load is immediately after startup. Alternator is already working hard to recharge the battery from starting and then with extra drain from heating elements... blah blah blah.... you already know that.

I'm convinced it's an accessory if not actually the alternator.

If you hold the gas pedal all the way to the floor before/while you're cranking it, the injectors will never fire. Flood-clearing mode. I bet if you crank it for 10 seconds (limit per the manual) even when it's warm you'll be able to replicate the sound.