Theist: Homosexuality is a sin because God says so.
Atheist: Interesting. What evidence can you show me that this supernatural agent exists?
Theist: :idunno:
Atheist: So why should I take this law from this supposed supernatural agent seriously?
Theist: Because it's in the bible!
Atheist: Dude, you have to demonstrate that this supernatural agent exists first.
Oh, good, a straw man! Can I play too?
Theist: Demonstrate to me that I am not legally allowed to hold an opinion based on a belief in an entity I cannot prove exists and the rules I think that entity promulgated.
Atheist: You are free to believe anything you like, of course. You're just wrong.
Theist: Then this is what I choose to believe, whether you think it is wrong or not. Demonstrate to me that I cannot vote my conscience based on my (wrong to you) beliefs.
Atheist: You are free to vote as you wish, of course. It's just illogical that you would do so.
Theist: Then we have nothing to discuss, really. I'll believe as I wish and vote as I wish.
Atheist: But your opinions are based on illogic!
Theist: So what? Freedom means I don't have to adhere to your logic, nor you to mine. But we all have to obey the law, and we vote for the society we want to live in, regardless of our reasons for wanting it that way.
We can play this game all day. At the end of it, though, is this. We live (in the USA) in a society that is secular, but within a framework of laws enacted by men and women who vote as they believe society should be ordered. Their religious beliefs or lack of same can and do influence their vision of the society they want to live in. If a certain percentage of religious believers of one sort gain secular authority, expect the rules of that society to reflect those beliefs to the extent not prohibited by the Constitution. That is the way our secular society runs. It is not run by the rules of logic, or by the rules of science, or by what one can and cannot prove to be true. It is run by people who hold beliefs, some logical and some not.
Your recourse is to work to change those laws by education of others and by voting. Arguing that their beliefs are based on a non-existent being is a non-starter. You won't convince them, and nothing forces the law to disregard the opinions of the religious voter.
Point out that religion is illogical, and based on the existence of a being that cannot be proven to exist? Great fun, I suppose. The point? I don't see one. People will not suddenly clap their hands to their heads and say
"Oh my goodness! I've been hoodwinked!" Judges will not strike down laws because the people who voted for them were religious and not atheists.
But look at the bright side. When atheists are the majority, then society will tend to be ordered more to your liking.