I think the following system, which I believe is more or less as they have in France, makes the most sense.

Marriage is legally a secular state institution. Anyone who wants their marriage to be legally recognised must have a state service. That covers all legal issues, such as establishing next of kin, inheritance rights, etc.

If a couple are religious they are at liberty to have a religious wedding too. As the state already legally recognises their marriage, the couple is entirely free to determine the nature of their religious ceremony; it can be conducted anywhere, and be as simple or extravagant as they wish.

This freedom means it is reasonable that religious organisations should not be obliged to conduct a marriage service for anyone they do not consider compatible with their faith. This should not unduly affect anyone wishing to get married, for they are entirely free to establish their own "church" that is entirely compatible with their beliefs. They have no need to impose themselves upon an organisation that does not entirely agree with their beliefs.

Everyone's a winner!