There seems to be a general misunderstanding about how VAT works. Not having VAT on school fees is not a tax break for the schools. The school doesn't incur the VAT, the people paying the fees do. A school, in simple terms, would merely collect the VAT from the parents and then pay it over to HMRC.
At the moment, private schools are not able to reclaim VAT on purchases. State schools however, through some special VAT legislation, can reclaim their input VAT, despite education being an exempt supply.
So in fact it is state schools that are currently enjoying a "VAT tax break". This "tax break" is not available to other entities that make VAT-exempt supplies, whether they are private schools or not.
All things being equal, if a private school were to simply add the 20% VAT onto its fees (as opposed to reducing the fees slightly and then applying the 20% VAT), then they would actually benefit from Labour's proposed change in VAT legislation, because they would be able reclaim all the VAT that they currently can't.