Nominal Catholics are often worse than professed non-Catholics, so I’m not sure that such a policy would really solve anything.
The policy would have to include eliminating employees who teach, speak, or act contrary to the Faith they’ve professed. (My original post specified “without dissent.”) People who say they believe something, but don’t, are dishonest, and should not be in positions where they can harm others.
I’m not saying this is a great answer. However, I do not see how one can have a functioning religious institution - grade school, college, hospital, charity - under the current conditions.