I've experienced the same problem with automatic petrol stations near TLS not taking UK cards, but on a recent (July) trip I discovered that the first petrol station beyond the airport (stay on the dual carriageway after the airport exit) accepted my UK Visa card. It's then no problem to go to the next exit, turn around and get back to the airport. No guarantee this will work for you - as our US friends like to say, your mileage may vary!

As for Avis itself - why not call them and ask, I imagine this issue comes up frequently.