it is lawful for the client to pay you the direct costs of your flight on behalf of your employer
I don't think that's right as "valuable consideration" does not have to pass from the client or customer to me, the pilot. All that has to happen for the flight to be aerial work is for the consideration to be "given or promised". This corresponds with contract law, where A can contract with B for a consideration which passes from B to C rather than to A.