If the dealer has e-mails, or other paper work from you, were you agreed to the higher price. That I would guess, would complicate the matter some. If on the other hand you have e-mails from them at the lower price, that would be good for you. Don't see how they would win if that is the case
That said, contact an attorney, The good ones are probably $250 per at least. Let them handle it, if that is the route you choose.
Dealer maybe crooked and just bluffing, an attorney may well back them down. Or could just be an honest mistake on their part. Hard to know.
If it were me, I would try and do what is right from both a moral and money point of view. Only you know what that is, in this case.
My 2 cents