Why on earth should there be a cap on private school fees? They are there to serve a market, and do so at different levels.
So what if some (probably most, actually) hard working professionals cannot afford fees? It isn't a right, or compulsory to send kids to private schools.
Many schools cater for the very wealthy, others less so. All will tailor what they offer, and at what price, to their market.
I see no more reason for the state to interfere with the pricing of a private sector. Should there be a cap on the cost of wedding dresses, or cars, or any other commodity?
We all benefit from state education. Instead of asking yourselves how much private educating parents save the state, ask how much those very parents would have to pay their employees if the employees had to cover the cost of private education. How their business model would need to change if there was no state health service and employers had to cover full medical insurance for each and every employee.
The benefits go way beyond the price of education to your individual child.