Accidents happen, it is inevitable. You are only human
Going with a billing system is a great way to automate your business, allowing you to become more efficient, etc.
If there is anything that went wrong in the process, like billing messing up. Just refund the customer (if it was a charge error) and apologize about the error.
Companies big and small make errors, as long you do the right thing and fix whatever error happened then everything should be fine.