You are wrong. No quoted rate is ever the PAYG rate, it's always the charge rate of a company. Otherwise there always will be someone willing to work for minimum wage which will be the same rate across the board.
Lots of things I am wrong about & really were splitting hairs.
He is not putting in a quote to or as a third party, if he were to invoice, I would suggest the lower rate as he was acting as the cleaner, if he were to charge what a third party would quote, I reckon he would come unstuck badly if was checked up on, first thing I would ask is for the companies records, insurance and books......his answer would be ???
If he were to charge a hourly rate that an employee would get, it would look more reasonable, but again, the rules are the rules, they may ping him for that, I do not know.
He has no legit invoice, the deed has been done, I suggested letting it go, even if the rate was 1000 dollars an hour, it is done. This is the crux of the matter really & the rate would not change my suggestion of letting it go.