After reading the whole thread, have to say OP that YABU
You were offered the option of a school which could take all 3 children. You turned it down because you thought your children wouldn't be able to make local friends.
You now have 2 children at one school, and one at another. But you don't want the 7yo to go the school where the majority of your children are, as the 7yo school is better, so you aren't that interested in getting them a place.
I think from what you said that the 4yo can't go to the 7yo's school anyway as they don't have an infants (correct me if I'm wrong). So you can't move the 4yo to the 7yo's school.
You have already appealed in order to get the 10yo into the 4yo's school and got the school to go over the class size limit in order to accept them - which affects the school and other pupils even if only minimally.
You can't pay for breakfast club.
You want the schools to make allowances for your child(ren) being late.
You seem to want everything the way you want it and expect the school to make accommodations rather than you doing so. You wanted certain schools rather than the one too far away - the price you're having to pay is that they're not all in the same one. If that is something that is not practical for you then imo you shouldn't have accepted it.
It's selfish to expect your child(ren) to suffer the humiliation of being late and the school to have to put up with it because of a choice that you made. It may have been a limited choice, but it's still a choice.
You've got the LA to bend rules already and accept your year 6 child - fair enough - but now you've got the schools you wanted, it's time for you to start adapting to the system and finding a way to get all your children to school on time.