It's been a while since I bought a car...but I used to always buy my cars on credit cards.
The bigger question to this thread is not "What will dealerships allow?" but "What do the new regulations require them to do?"
In the past, even if they whined, any merchant that accepted a credit card for any transaction had to accept the card for all transactions. They could not set a minimum (as many small mom-and-pop restaurants do) and they could not set a maximum (as, in this case, car dealers tried to do). Lastly, they could not raise the sales prices commensurate to the fee nor could they charge the fee to the consumer. When presented with this info, every dealer capitulated and allowed me to use my card.
But a few years ago, when Congress (damn them) enacted some "enhancements you will like" there also was language that supposedly changed the conditions I set forth in the previous paragraph. But, since I am not currently in the market for a new car, I have not looked into it anymore.
Does anyone know "for real" what is required and what is exempted? Only then will we know what's going on ... and only then do we know what might be a bargaining chip.