tl;dr - There can be no freedom without responsibility.
Considering that the government is supplying these corporations with limited financial liability, I would argue that being covered by that government umbrella makes any such covered organization, by necessity, limited in rights due to that government associated limited liability. Thus this "religious freedom" argument is as much a violation of the first amendment as if the "religious freedom" of a bureaucrat to require you to share their religion in order to provide you with a service.
I would go further and argue that the ability to grant limits on liability is an explicit violation of the equal protection clause of the Constitution. It's either that or the Government is partially extending its own immunity to the entity, limiting their rights in doing so.
Long story short, the ability to grant limited liability should require a new Constitutional amendment and any person operating under that limited liability umbrella must have, due to the limited responsibility, limited rights. In the absence of an amendment spelling out those limits, the only way to not violate equal protection is to consider persons operating under limited liability to be implicitly operating on behalf of the government.