Al,
I am guessing that you are now lodging with someone (parents?) on a temporary basis whilst you sort yourself out and have had your new card sent there. The letter which came with the card should provide one of the proofs of identity needed, so maybe just get something else sent there. I am sure you can find a reason for someone to write to you there!!
An alternative might be to get someone else to rent the car, naming you as a second driver. That way you would only have to show a licence as they would be legally responsible for the charges. Being the trustworthy soul that you are I am sure someone would assist you.
One other point. I think it is a requirement to advise the DVLA of your current address and so you should consider letting them know that you are no longer at that old address and are temporarily where you are now. I think you can do this online. This only becomes important if you have to show the police where you are living.
Hope this helps,
David
