In any of the high-profile cases, the issue was never merely selling merchandise to gays. I think all of them, at one point, said gay people were perfectly welcome in their stores to buy goods "off the rack."
The issue was freedom of expression--that on First Amendment grounds, businesses cannot be compelled to produce custom-made merchandise or services that go against their conscience, such as cakes or floral arrangements for same-sex "weddings," T-shirts for so-called Pride Month, and the like. The owners of the businesses do not want their names associated with statements or causes they don't support.
It seems to me that the difference between that and the case in the OP, is that hairstyling isn't an expressive art. About the best you can do in terms of messaging is dye your hair in the colours of a pride flag, or something. And so the refusal of the salon to do business with transgender people isn't really on free-speech grounds. It's more like good old-fashioned discrimination. If it goes to court, I don't think it will fly.