If he never made cakes and believed all cakes were evil, it would be wrong to try to ask or force him to make cake. But when he makes cakes for a living... no, it is not wrong to seek to patronise his business.
It is not necessary for the baker to believe all cakes were evil for it to be wrong to ask or force him to make cake, any more than it is necessary to believe all meat is evil in order for it to be wrong to ask or induce him to at meat, for while you consistently ignore critical details, if one only believes meat sacrificed to idols is evil, then it would be wrong to ask or induce him to eat this meat.
Thus ifI say it violates my conscience to rent apartments to minorities or offer them jobs I can discriminate all I want. After all, it is my standard and who are you to tell them otherwise.
I think this is a pretty weak standard for being able to break the law.
But this reinforces the point that "I believe something therefore I get to break the law" is a pretty terrible standard for allowing discrimination in arbitrary cases. People believe all sorts of things - where do we draw the line? Do we allow everyone to do anything they want under the guise of "but I believe"? Or do we get the government involved in vetting religious beliefs?