Wow, I'm genuinely shocked about this. What a massive PR own goal.
If that ever happened to me I'd definitely contest the fine as a matter of principle, and if necessary, take my chances in court. I think McDonald's legal position is pretty shaky to say the least. For a start, how can they prove you stayed too long? Who decides how long too long is anyway? And who gets to decide the size of the fine? Even if it was justified (which it's clearly not) it seems excessive to me.
Also, what the hell are the DVLA doing giving out personal confidential information to a private company?

This is just wrong on so many levels.