Hi Slim,
This happened to me, but because of the delay, often in excess of 5 weeks when waiting for your car tax renewal, I find it better paying the full amount online, and then requesting the refund from them, because of how long it takes. You're fine asking for a letter, I've done this many times - explained that I needed a duplicate copy for DVLA and I received it within days.
Hope this information helps - remember to send them your original V5C form, or they'll reject your claim. They do post it back to you.
Chris.
Nothing on this board constitutes legal advice - always consult a professional about specific problems 
