300W isn't very much; fan + rear de-mister would probably draw around that...

I thought so, too.

How many amperes can the alternator deliver, max?

60, on paper. I'll have to check if mine still can do that.

To me it sounds as is the alternator isn't delivering; once started the alternator should provide all the current you need, as well as charge the battery. Did you have the battery on a charger overnight?

No, I figured I'd wait until the next morning, as I have experienced batteries to recover a little over night.

What are the voltages across the battery 1) at rest (car off) 2) car idling and 3) car at typical running RPMs?

The voltmeter I have installed (factory harness) says 1) 11.5V (ign. on), 2) and 3) are just under 14V, but drop about 1.5-2V with empeg and headlights.
I'm going to keep monitoring the behavior and measure accross terminals for a few days. I still think some more driving will restore the basically brand new battery in a few days and this weird behavior will subside. If it doesn't, I'll measure the max amps the alternator can deliver and replace it, if I have to.
Thanks for all your help, guys! Keep the suggestions coming!