Update: I just spoke to an Avis UK representative on the phone, and they told me that it is technically not against the terms and conditions to continue to book on the Avis UK site with my UK account.

I'm quite surprised by this and wish I could find it in writing somewhere. I haven't actually seen anything definitive either way in writing.

Further edit: The above was confirmed by another Avis customer service rep and also I was told by the first agent that his supervisor was basically telling him that it was fine to continue using the UK site as long as my UK license is valid and so on.