If your credit card has a high enough limit, then I would first call them to make sure they temporarily raise the daily limit for the single transaction.
Then calculate what the rewards points would be worth.
Negotiate the price of the car after trade in first, telling them that you are going to pay for it on the spot.
Then once you have the agreed upon price and trade in value in writing, tell them you want to charge the entire amount to your card.
Then if the dealer gives you a hard time about doing it, simply tell them they allow you to charge it all (with no extra charges for using a credit card) or give you the value of the rewards points off the transaction or you are going to go to another dealer who will let you do this. The car industry is so hungry for business right now, they should be more than willing to do this to keep a customer. Especially if you are buying a car that will cost $20K after trade in, as these cars temd to have a higher profit margin, since they are over priced (basically you are either paying for the name or a premium price for features that are nice, but in reality actually cost very little to manufacture and install) I am not saying its not worth paying for those features, just realize that the features cost A LOT less than what the dealer is charging for them.
On that note, there are now several websites that will tell you what the dealer's costs are before you go, I strongly recommend using them. There are also several websites where you can tell it what kind of car you are looking for and your zip code, as well as what your trade in is and it will find a dealer in your area and give you an offer that includes all taxes, dealer fees, etc. Even if you don't use those dealers, I would recommend having an offer in hand when you go to the dealer. We got our car for about $500 less than the offer by negotiating a bit.
To that end when negotiating, make sure you agreed upon price includes all of these fees as well.
Another option may be eBay motors, especially if you are looking for a brand new car. You would need to do your trade in on its own in many cases, but there are many dealers who will buy your car, even if you aren't buying one of theirs now.