Catholics? Separation of church and sate? Why?
If you run a charity such as a hospital, school, clinic or social service and you accept funds to run your operation from the federal government, the state government or insurance companies you have to expect to abide by their rules. If you want not to, you don't take their funds.
Seems as simple as that to me, a retired R.N. (Remember "he who has the gold" rules) Or, they could just use money from the feds to pay the insurance they oppose and then it would not be "them" paying it.
By accepting money from anyone but their own sources -- i.e. church's contributions, individual donors or other "no strings attached" donations, you are "free of the state" and can do as you please, within the law of course.
This is how the Shriners run their hospitals. They are a true charity.