Hello I am a solicitor with over 15 years experience. I will try to help you with this.
Am I right in thinking that you have not actually suffered any losses as result of the dealership's mistake?
If, as seems to be the case you have not suffered any loss (indeed you suggest a way in which you may have been saved money and possibly penalty points!) then there is no claim to be made in law. The point of compensation is to compensate you for losses suffered. There is of course no reason why you can not write to the dealership expressing your concern and explaining how upsetting you have found the whole situation of being exposed for a long period of time to potential difficulties with the law. You could take the opportunity to ask if there is trade body you can complain to if you want.
This would of course just be trying it on but if they care about their reputation they may offer something.
I old think a letter explaining their error might be helpful just in case something comes up. I don't think there are any magic phrases to use just explain the history of it, thank them for fixing the plate but then tell them how worrying you find it that you have been driving round displaying the wrong plates. That this has left you vulnerable to prosecution (hence you want the letter) and continues to cause you worry and distress. Invite them to suggest ways they can make up for this rather than coming up with a figure (or other benefit like free servicing) yourself.