The legal foundations of the privilege is given in the 1st Am of the US Constituion and affirmed in the Civil Rights Act of 1964 which excludes religious and some private associations (e.g. "clubs") from scrutiny of equal protection compliance through charter or membership.
wiki lists 14 of 17 (according to Amnesty) state legislatures that've adopted conscience clauses that, one way or another, circumscribe medical benefits. Americans lurv their children but hate their women.
See for example Pharmacists for Life if you dare. Diversity is the key to economic and political evolution.
The argument that the Ladele case tried to put forward was that she shouldn't be forced to go against her beliefs and do something that she felt was wrong (ie a conscience clause should apply for her). I'm very pleased that the ruling didn't go her way. And yet those who protect the rights of same sex couples not to be subject to prejudice and discrimination are the intolerant ones... Ad astra per aspera
So refusing to officiate civil partnerships cannot be based on religion be cause she is not joining together under the sanction of god. Heck she isn't even marrying.
She's just a bigot who thinks mumbling to the spirit in the sky (be doo da da da da da) gives her a pass on humanity. keep to the Fen Causeway