I'm not sure I would consider this just a flat rate problem.
Would the customer have felt any better about it if they were qouted an after hours hourly rate, then after the plumber came out did the diagnosis and made the repairs they were handed the bill after the repairs were done for the same amount?
At least with the flat rate the customer was quoted the exact price before the repairs were made. With the T&M hourly rate charge the customer doesn't know the price until after the repair is made and they're given the invoice.
If the customer didn't want to pay the $100 for the diagnostics they could have just paid the $79 for the plumber to do nothing. That is if you consider driving out to the customer's home at 8:00pm on a friday night as doing nothing.
