Not sure if there are local variations of the rules, but
according to BA,you get the additional driver, and collect Avios (and I believe the other benefits) simply by looking with your BAEC number. Using Avios or cash makes no difference.
Executive Club members are eligible for one free additional driver on every qualifying rental, worldwide.
Simply log in to your Executive Club account and make your reservation as normal. You do not need to request an additional driver at time of booking.
Simply request an additional driver to be added when you collect your car. The additional driver must meet standard Avis age and driver requirements for that rental location.