It's summed up nicely in the first antiphon for Morning Prayer in my Breviary today(Wednesday of Holy Week, for future reference).
"Evil men said: Let us make the just man suffer, he sets himself against our way of life."
It is, quite simply, a war of mutually exclusive ideas.
Seriously, in this day and age, are you completely unable to get a cake made by someone else? You HAVE to force a Christian to go against their principles?
I get that their side and mine disagree. I even get that there are some Christians who would absolutely love to force gays to act straight.
Here's my solution. If it's not biblical, let me know.
God gave us Free Will. Some people abuse that Free Will by going against His Will, and do so knowingly. God freely respects that choice. I suggest we do the same.
This is not to say we can't be like John the Baptist warning people of the consequences of their decisions. Not condemning, mind, warning. And for the love of all that's Holy, don't go full Westboro. Never go full Westboro. DO NOT BE HATEFUL. Witness to the Truth, in Love. Witnessing to the Truth in Love was how Jesus got the despised sinners, e.g. tax collectors and prostitutes, of His day to repent.
Thing is, just like John the Baptist, we risk, when we do so, losing our lives for the sake of the Truth. John called out Herod and his wife for incest. He lost his head for it. Jesus ticked of the Religious Authorities of his day who were more interested in secular power than in doing the Will of God, and got Crucified. This is what it means to be a Disciple of Christ. Your reward is not in this life. Remember that.