I'm sorry to hear you've made a mistake. Having read the contract of sale we signed, I do not see any remedy to your current situation. However, I'm interested in making sure I do not take advantage of you. As such, I'll return the vehicle, you can return my money plus the bank fees I paid for the cashiers check, tax, title, and registration, and I will look at buying a vehicle from another dealership.
This seems to be the most fair resolution. If I were to pay for your mistake at a price I did not agree to, it would not be fair to me. If you were to allow this vehicle to go to me at the price we agreed to, it wouldn't be fair to you. If I were to return the car and begin negotiations again, or find a different car in your lot, it would be difficult for us to know that you were not going to make a similar mistake again.
At this point I consider the sale final, but if you'd prefer to have the vehicle back as-is, returning to us the money we gave you as well as the additional costs incurred by the sale, then we will do so in order to set things right.
Chances are good you will see them back down. Perhaps they will just cut the additional payment in half, and say, "Well, it's our mistake, so we will eat half the cost," or similar, but this is merely another way to get you to pay more money.
Stand firm. "I appreciate the thought, but I cannot accept that offer. When will you have payment ready so we can return the car?"
If you are firm that the only two solutions is to keep the car, or return it for a full refund plus associated costs, I'd guess they'd rather you keep the car - trust me, they still made a profit - but if they decide to have it returned, do so and make sure they pay you in full plus other costs. Bring all your receipts, etc and don't hand over the keys until you have the check in hand.
Then go, gladly, to another dealership that doesn't abuse its customers so badly.
If you do end up keeping the car, don't plan on going back to that dealership. Use another dealership for warranty work, and find a good mechanic for non-warranty work.
Note that this solution isn't legally required in most jurisdictions. Read your contract and all documentation they provided at the time of sale to be sure, but it's unlikely that you are legally required to make another payment for a vehicle after the sale is finalized. Even if they haven't cashed the check, the sale has already been finalized.
What this solution does, though, is put you back in the driver's seat in negotiating. Right now they are treating it as though you owe them something, and thus you might feel an obligation toward them. Re-asserting your relationship with them as a customer rather than a debtor is very important regardless of how you proceed. You aren't legally culpable, and so making sure they understand you aren't will ultimately help you.
Further, dealerships operate on negotiation. The primary power the customer has in the dealership is the power to walk away from a deal.
They've set the situation up as though you no longer have the power to walk away. They didn't threaten with re-possession because they can't - the sale is final. They presented as a one-path situation - you pay. Period.
You do have many options, though, and they are very familiar with the "walk away" option. Present that as your chosen option - either they stick with the original deal, or you walk away - and they will have to look at getting another car off the lot (which is often more important than making a profit for a dealership) or selling a slightly used car. If they've correctly pushed the title transfer through (or you, if that's your task in your state) then your brief ownership will show up on carfax and similar reports, and instantly reduces the car's worth. Having the title transfer immediately back to the dealership doesn't look good to future buyers.
So the dealership doesn't want the car back. They are just trying to extract more money, and probably illegally, depending on the laws in your jurisdiction.
Reassert your position as customer, and decide now that you'll be fine if you have to return it and walk away. Then when you communicate that to them, chances are good they'll simply cave and let the sale stand as-is.