Spur of the moment attempt:
Imagine a card player who sits down to play poker with you. You explain the house rules to them, and they agree to these rules. They then proceed to win a number of hands, while using the house rules to their advantage (deuces wild, high-low straights, the whole works, whatever it is that your particular card group buys into). A few hands further on, they go in big on a hand, and lose to someone else who is utilising one of the house rules. They object to the use of the house rule in this case, and refuse to give up the money that they have lost, leaving the game in a huff.
The religious defender of relativism about ethics does not, standardly, want to abandon reason in favour of religion. Rather, they want to be able to use both reasons and religion when it suits them, while reserving the right to claim not to be bound by reason when it clashes with their religious convictions. If you engage with them, while allowing them to make this move, you can only ever lose in the long run, because they are simply not acting in good faith. They are purporting or pretending to be bound by a series of conventions (reason), which they are willing to sacrifice (and to deny ever having been bound by), when it suits them.
The religious objection that you are ‘privileging reason’ without good authority, collapses as it purports a false dichotomy between choosing to follow religion or reason, whereas the actual goal is to follow both when it suits them, and religion alone when it does not suit to follow both.