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

How many amperes can the alternator deliver, max?

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?

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

/Michael