People have used sincere religious arguments against interracial relationships for decades if not centuries. The reason it's not invoked now is because we've had a couple generations where the law of the land was obviously morally superior to the scriptures, to the point it's not seriously debated anymore.
There are absolutely sincere people whose religious beliefs are that interracial marriage is morally wrong[1].
> Reagan says many black athletic stars choose white wives in a willful attempt to make their offspring lighter, challenging God’s plan. “He don’t want them to be like him, so he’ll marry another. … It’s another defiance of God’s law, it’s a worldly way.” And the pastor condemns fellow ministers who perform interracial marriages. “Some of the men in pulpits should have a pantywaist instead of a preacher coat on!”
[1] https://www.splcenter.org/hatewatch/2014/02/19/tennessee-pas...
What's actually in scripture has very little to do with the religious beliefs many profess.
> we wouldn't be as conflicted it it was an interracial couple that he was denying services to
I don't believe you would be able to make a "sincere religious conviction" argument against it - I'm not aware of any mainstream religions that prohibit interracial marriage as part of their doctrine (as opposed to an epiphenomenon of the cultural practices of the people who make up the group).