Does the RAF reimburse the cost of a car wash as well as the cost of the refuel? If so, then there is not a great problem in putting the car through the automatic car wash at the garage at the same time as refuelling it.
Otherwise, was there an automatic car wash available at MT? If not, then did MT provide the facilities to wash the car, including the changing room for the Sqn Ldr to change into overalls prior to washing the car? If MT do not provide the tools and the facilities to enable the car washer to keep his uniform clean, then they cannot expect the order to be complied with.