Forget about Muslim. What if the business owner (a barber, or a tailor) is simply not qualified to serve a woman? If I am a tailor who specializes in man's work pants, how can I take an order for an evening dress? Does the law somehow magically make me qualified to make dresses? Or does the law automatically forbid me from doing my job just because I haven't learned how to do some other job that I'm not interested in?
In this case the Muslim barber is right, as long as one supports freedom of religion. Perhaps if he is an employee, the owner may have a problem with him; but if he is a partner, or acts with agreement of the owner, it's their business. It may be that they are losing orders from female patrons; but it's their problem; and if they are alive and well since 1925 then I'd say they have enough customers.
I also don't know why the journalist just had to point out that the female customer was LGBT. It has nothing to do with the issue at hand, and I'm sure the barber never inquired about that little detail.
bump
“I also don’t know why the journalist just had to point out that the female customer was LGBT. It has nothing to do with the issue at hand,...”
It _may_ have been relevant in that the womon in question was a “butch” and wanted a “masculine” haircut. Who better than a men’s barber[1] for the job.
[1] Ironically, the word “barber” is from the latin word for “beard” which is feminine. {irrelevant funfact}