Well, what I think is that everyone should have a right to participate or not participate in speech. This means I have to support the bakery, because they could also prove it was the message itself they objected to, due to their beliefs. They weren't actually refusing to even provide a cake, just didn't want to ice it with the words saying "support gay marriage."
Regarding the billboard or other private companies refusing to publish gender-critical statements, that means I have to technically agree with their rights to do so. However, I think the case of the dictionary definition of woman is different than a religiously-grounded refusal to produce a message specifically to support to gay marriage. The onus, I think, is on the company itself to justify their objection to the actual message itself, i.e. the dictionary. Which would probably be impossible, really. Then they would have to look at their own policies. Are they applying the same standard to everyone? Would they let another group put up a poster with a different dictionary definition of word? Would they let certain religious minorities share their beliefs on a poster, even though this could offend other minorities or atheists? Would they let political parties put up messages even if they are controversial or offend other voters?
Or is this a specific systematic refusal of service to women who believe in biology - and if so - how can they possibly be justified in gagging only such women, if they allow other groups to advertise? What is their "rulebook?" Like how Christians can point to Biblical texts?
If they are anti-women, then at least they should be honest about it, explain why it is only women having free opinions that they find problematic, and attempt to justify why they think the content (in this case the dictionary) is now fundamentally against their beliefs. In such cases, these outlets likely do not have a leg to stand on, unless they publically state that they now find defining certain words a hateful act. And then we can all laugh at them for having gone absolutely stark-raving MAD.
I think the issue with free speech here is the statement itself (not the group making it), and whether you are being made to produce it yourself (as in the case of the bakers, i.e. compelled speech). I would find it highly philosophically objectionable to make a cake that says "Repeat after us: trans women are biological females," "TERFs are trash" or "Smash the cotton ceiling." Therefore, it is also in my interest to defend the rights of these bakers not to participate in producing messages they fundamentally do not agree with. I am sure the bakers would have iced the cake with something like "Happy Birthday."