You can immediately pay off your credit card balance if you so desire and credit cards (at least American ones) don't charge interest initially. So if you automatically have your credit card debit from your checking account each month there is literally no cost to you to use a credit card and only upside.

The merchant is already pricing in credit card fees into their prices, so you should go with the payment instrument that provides you the best protection and other perks — otherwise you are leaving value on the table.