for consumers it is as John said, counter-intuitive. For commercial customers, most banks charge a set fee for debit or credit transactions posted, either per item or a monthy minimum that becomes per item once the per-item exceeds the minimum threshold.
Now, in the scenario above, it sounds like you could be setting up an origination where your bank generates the ACH entry to debit your customer and provide the credit to the other bank - if you are talking origination, then all banks i'm aware of charge a fee for this. Assuming the customer uses some sort of cash management tool or product to create and send the payment.
When life gives you lemons...squirt the juice in your tormentors eyes...that stuff burns!