As far as I see it, if they're getting special financial consideration from the government, it's as good as getting government funding. Government-funded institutions have to respect the Constitution, and so should they.
Frankly, I would question the priorities of any institution that would rather shut down hospitals than respect its members' rights and privacy.
Oh, why don't you go pick on the Amish or something? They have rules and restrictions of "members' rights" up the wazoo, and I believe their churches are as tax-exempt as any Catholic one.