In my view if the school only wants to hire Catholic teachers and only teach Catholic children then they should not be eligible for any State Revenue. And that goes for any other religion too.
If they want to be a really private school then don't ask for any public money.
It's important to remember the Catholic Church built the first and best schools and hospitals in this country, and that they're still going strong through popular demand. This came about because people with a common belief system and commitment to improving their community worked together. People with a random hotch potch of half baked ideas, as are likely to be found in your average government school can achieve very little.
As someone commented when well known atheist writer Philip Adams got carted off to St Vinnies during a medical emergency. "Why didn't he go to the atheist hospital? Oh that's right there isn't one"
I support the same principal for Jewish and Muslim schools, they should also have the right to choose teachers based on their religion. I would much prefer to send my child to one of their schools in preference to a godless school, where there is no common morality, kids tell the teachers to f--- off, and promiscuity is OK, in fact anything
Forcing democracy upon religious institutions means there can be no religious freedom, in fact democracy is a tyranny in the hands of the myopic.