If you can't do it before the end of term, can you maybe do it throughout the first week of September?
I have to do one week (5 days) Primary School Experience and will be doing mine from 5th Sept and start uni on the 12th.
...so if the schools you have contacted are using schools kids work experience as an excuse...perhaps suggest September - surely that won't class with work experience?
Also, maybe expand your search and be willing to travel a bit (of course this depends on transport links where you are).
I live in London and will be doing my PSE at a school that will involve me leaving my house at 6.30am and driving for an hour and a half!