... Avis in particular always seems to ask me whether the address on the licence is the one I want to put on the rental contract or whether I want to use a different one. (I used to do so when I lived in Paris and still had my UK address on my licence.)

What's probably more important is that the address you give them matches the one on your credit card; if it doesn't, presumably the credit card check will fail when you go and pick the car up.