So is the bottom line that you want to make independent school affordable by buying part time?
If I were a prep school Head I would be wary of setting this as a precedent as I would think I was opening myself up to families wanting to buy one or two days a week as a sort of tutoring service on top of state school.
It's a totally different scenario, but I have actually taught several classes in which there were part time children. It has never worked well. It makes getting and keeping the class settled into a rhythm and working as a team much harder. It is also hard socially for the part time child as they are constantly having to re-establish themselves into friendship groups when they have missed what happened in break time the day before etc. However much pre-teaching/ re-capping you do for the child you still end up making references to things in lessons that happened when they were not there which makes it difficult for them to feel part of the group.