I' have all my autopayments set up on a specific card that never leaves a drawer in my house.

End result is that when the burner card gets compromised, as it does several times a year at gas pumps and such, I toss it out, get a new one and it's not all that painful. The autopayment card has yet to be compromised.

I wish it were easier to get temporary card numbers without giving someone else access to my purchase data.