Can I pay to sit the higher gcse Watch
I presume you're not taking it before next Summer to the tier doesn't have to be finalised until the end of the Spring term. Forget about trying to take it privately. Work hard and get the grades to justify your school entering you for the Higher tier exams.
Lots of things are possible but it doesn't mean they're going to happen. If the school is convinced the OP's best option is to take Foundation tier then why would they agree to this? It isn't going to happen.