I've had to do this before.

I was on a transatlantic economy ticket. Got caught up in weather problems on arrival at LHR, and was asked to take a train.

I travelled first class, and sent BA a photocopy of the ticket, with a covering letter referencing the walkup economy fare, and saying that while I have enclosed a first class ticket, I realise only economy may be refunded.

A reply to the letter was never received, however, they did send a cheque for the full first class ticket, which was unexpected (and unnecessary).