I don't see in any way that there is a conflict, because a hospital, charity, etc, that bears a religious name or has ties to a religion is not a church. These places or organizations are supposed to be for the benefit of society as a whole and not strictly for the pushing of ideas on others. If a hospital, organization only wants to aid its members and not society, this would be a different story. These places are open to the public, not just its' members.

They didn't order any religion to preach the use of birth control, churches are exempt. If Catholics don't like helping those outside their religion, then their hospitals should be deemed as private hospitals and not except money from any type of insurance that doesn't fit their religion.

Room for thought, does the Catholic church feed only the hungry that doesn't steal, kill, commit adultery, etc.

If these organizations do not want to follow these guidelines, then they don't have to receive government assistance. If they receive government assistance then they have to conform to the guidelines, especially when contraceptives have a proven health benefit.

If they want to maintain their anti-contraceptive stance, by all means go private.