They don't put on extra fees, but they would not give you the discounted local fees either.
A private school normally quotes a price for Domestic Students, ie: after the government have paid their portion for the student.
eg:
Private school may want $12,000 for the year.
The Government may contribute $6,000 for the local student, so the school only charges $6,000 to the student.
However, for non local students, (ie: No Government subsidy for the school), they need to charge the student the full $12,000.
There have been cases where the $6,000 gets quoted, but it goes up to $12,000 when the school realise there is no subsidy, after the visa gets submitted.
It might be worth checking with a local private school, if they even have vacancies. Ours had a 2 year waiting list, and would offer no discounts.
But I am aware of one that had plenty of vacancies, and allowed a local price for a non local. (ie: They accepted a reduced payment just to fill a place.)