Pressed for time, but while in a land of freedom, no one should be compelled to do business with anyone, even if merely on a whim even, etc. Even racism isn't overcome by forcing people, but is instead overcome by love and words and deeds in reality.
But....as Christians we know we are to continue loving people even if they are sinners (or even if we make guesses that way, a risky thing to do btw).
If we stopped loving people if they are sinners, we'd only love a few old people, and none more, here on Earth.
So when we are obeying Christ our Lord and loving a sinner, how might we interact with them?
Answer: in a truly friendly way, we should tell them the Good News of Christ that they might be saved. This can be done even in only a few words in some ways, at times.
Actually doing that. So while the baker may already be willing (it would appear from this article) to make them a cake without a theme he personally disapproves of, quite reasonably, he could bake them a generic cake without that theme and tell them the Good News. He might still get challenged, or possibly not, but even if he is challenged he will have done all the right things.
It's interesting to consider that in the scriptures there is nothing at all about gay marriage of course, not one bit.
And we do know we are not to add to scripture. It's a commandment in the OT.
What is in scripture is references to sexual immorality -- actual deeds or committing adultery in the heart. Not merely for instance a celibate love relationship (as an illustration). Often this issue is argued by people not seeming to realize the actual wordings in the scripture, about a particular detestable act that even male/female traditional married couples can do, and some do. So, they end up arguing against 'gay marriage', which isn't even a thing in the Bible at all. And, are they themselves free of every sexual sin? But all have sinned. All have fallen short. We all equally need Christ, you and I as much as anyone else.