In my experience (YMMV) if you flash your EC card when collecting and providing you have both licenses they will add the free driver how ever you made the booking.

I cannot seem to work out how Avis set their prices which seem to vary between Avis.com (with BA AWD), BAAvis.com etc. The last time I booked by phone the price was even lower than the live price showing on baavis.com.

I suspect the confusion isn't entirely accidental and doesn't hurt their bottom line - but does mean it pays to shop around!