No. Only use your credit card when you are money rich and time poor.
As you've noted putting the transaction through VISA, Mastercard (etc) will cost them ~$400 in transaction fees.
A bank cheque goes for ~$15.
Is an hour of your time worth $385/hr? Let's be generous and say that all-up it takes you an hour to physically get down to a bank branch during business hours. You pay your ~$15 for a bank cheque from the teller and take that to the dealer. Save ~$385.
Obviously, you only save that money if, when negotiating exactly what number to put on the cheque, you factor this in by pointing out that by getting a bank cheque you're saving the dealer $400 in transaction fees.
One strategy is to haggle hard without mentioning it until you agree to an exact figure and it seems to be all final. Then spring it and shave another 2% off the total. Alternatively, mention it early and they'll be factoring into the negotiation all along and allow you to drive them a little bit lower than when paying by CC.
(Beware that most AU banks regard an over-the-counter event as a sales opportunity so don't be caught unaware: their kind offer to help get a better interest rate, or credit card—or whatever it is—it is not necessarily you they're trying to help).