I have no children in private schools and no financial means of ever sending children to private schools and I am personally in favour of a strong system of state education.
Yet to me the question seems fairly simple.
Hypocrisy= failing to practise what you preach.
So in order to decide whether this cleric is hypocritical, we only need to answer one question: has he preached the superiority of the state system?
(would be most unlike any sermons ever preached by our vicar; they tend to be far less political than that)
Or if he has not, is there any indirect reason inherent in his cloth that means he must be assumed to support the state system?
(again, I do not see it; the Bible is rarely very clear on political questions)
Or if there is no reason why he must be against private schools because of his holy orders, is there anything inherent in his position as a school governor that means he must not be seen to support a private school? And if so, would this be any different from the situation of any other school governor who does not happen to have children at the school?
(again, I cannot see it. Imho he can have perfectly valid reasons for sending his own dcs to private school, but still wish to do a bit of good for a local state school- or be pressurised into doing so by virtue of his office)
Hypocrisy would only arise if he was actively fulminating against the independent school system from the pulpit.
Or, as suggested by zanzibar, if he sees bad things in the school he is governor and does not work hard to improve them.