Are either schools within a mile or so of home
If so, kids at that school could walk
most schools would not allow a year 3 child to walk. Unless the OP can get the year 3 child in, the problem is not solved, as the reception child still needs dropping off.
Sorry OP I had misunderstood that it was the year 3 in the other school.
OK, my plan would be this:
Write again to both schools. Ask for subsidised place in year 3 breakfast club, or that they will continue to be late.
State clearly what your long term goal would be.
I would want to year 3 child to move to the same school as the others.
Repeat to school, if a place could be found for year 3 in X school, the problem will be solved.
Strategically, I would make the reception/year 6 the late ones, and say to THAT school, the lateness will stop if you find a year 3 place, they have 2 late kids, v. one place to find!
I would check with LEA that you are still on the waiting list for that school for year 3. Then phone once a month and ask. It is very unlikely that there won;t be some movement in kids over the next 6 months. You may not be at no.1 on the waiting list, but once one child has gone, the school may be prepared to wriggle a bit (legally at the moment if you are no. 2, they cannot let you in unless they let no. 1 in too)
Hang on in there, something will shift, Keep putting it in writing that you can't be in two places at once and can't afford £1,000