Ultimately, the source of the conflict here is in holding simultaneously that health care is a right and that health coverage will be provided primarily by private employers. If you believe both of those things, then you have to coerce private employers into providing coverage that meets some kind of minimal standard.
In our world, where some health care services are deemed morally problematic by some significant private employers, that would mean coercing those employers to violate their consciences. (By the way: what’s special about the Catholic Church in this regard? Does the strictly Catholic sole proprietor of a national pizza franchise lack a conscience? Why is it okay to coerce him into providing services he deems immoral, but not okay to coerce a Catholic hospital?) If you don’t want that level of coercion, then either you need to give up on the idea that health care is a right, or you need to give up on the idea that health coverage will be provided primarily by private employers.
I vote for giving up the second idea.