I use my bank’s bill paying service to pay for bills. It is free and, in most cases, the payment is sent electronically directly to the biller.
In a couple of instances, I have had billers try to charge a fee to pay a bill using ACH. I am very careful to watch for these junk fees and avoid them if at all possible.
Can you open a separate, dedicated savings account to pay the bills and skip the credit card surcharge? Just add enough to the account that the bill issuer can not clean you out by “accidental” advanced payment.
I think it would have to be a checking account, but yes, you could open a separate account and use it just for paying bills. You would want to do it at a different financial institution.