Well here is my thinking on this:
1. It would seem beyond dispute that both you and the DVLA accept that the winning bid came from your device.
2. An auction bid is a legally binding contract.
3. If the DVLA refuse to back down on the contract (and they do not have to) then you will end up in court and will lose. Unless you can prove or demonstrate good reason why the contract should not be enforced. It does appear from the DVLA response that they believe that your bid was placed to stop another bidder from winning the auction? Why would they think this?
Your only option(s) are:
1) Tell them you will see them in court and hope that they back down.
2) Pay up and sell the plate on or keep it for your own use.
The courts can only rule on facts and not what is fair or not fair. Sad but true.
FYI: I went through the same process with a bidder on ebay who won a set of wheels from my auction. Having done so he then attempted to retract his winning bid stating that he didn't know that the wheels were black. This despite multiple photos and a description stating "BLACK". A check on his feedback showed he had a history of retracting winning bids. He now owns a set of BLACK wheels.
Bruce , while much of the above is true , if you read the DVLA manager's response to the OP "Therefore, had an error been made you would have had the opportunity to call and advise us."
According to the very first post in this thread that is exactly what the OP tried to do . Instead , the call handler , again according to the OP , shouted down the telephone at him ( unprofessional conduct ) and refused to listen to what he was saying . These calls are generally recorded , so the OP needs again to ask the manager to listen to the recording , backing everything up in writing , and citing the call recording as evidence should it come to court .
At the end of the day , the OP acknowledged there has been an error , which may have been his or may have been DVLA's ( his admission ) ; DVLA imply in their response above that if he had called to point out the error they could have sorted it out ; he claims to have tried to do just this , has escalated it to management .
I think I can see who is being fair and reasonable and who is being obstreperous and unreasonable - so , I imagine , would a Sheriff sitting on the bench .
There have been numerous publicised cases of DVLA trying to run roughshod over individuals regarding non declaration of SORN , and being sent home with their tails between their legs thanks to 'little people' who had a better understanding of the law than they did .
Courts don't like government agencies bullying private citizens .