I think the compromise has to happen in a slightly different place.

The problem is that a person's sexual orientation is an immutable characteristic, which means if they get married, it's not their choice whether it will be a "gay" or "straight" wedding.

What if there was only one bakery in town that made wedding cakes, and they refused to make the wedding cake because it was a "gay" wedding? Well then the person gets unequal access to a service that everybody else in town has, just because they are gay. And that's not good.

And there are a lot worse scenarios where the same precedent holds. For example, what if a hotel didn't let a gay couple spend the night, leaving them without lodging in a strange city?

I agree that businesses shouldn't be bullied to accept every client for every service. But they shouldn't be able to deny a request for reasons that the prospective customer has no control over. It has too much potential for oppressing people for things they can't control.