All my US colleagues were Ms by default, at least in speech in all-hands calls. In documentation, they were usually just firstname surname, and if was some sort of article, they'd usually just be referred to as surname after the first mention. So I would have expected more systems to include as an option.
Anyway. I asked my pharmacy. They said no, we can't and pointed me at my GP. That's NHS, not private, though. I think I had to get a hospital department to update it separately, too.
What I appear to be unable to change is pharmacy messages to firstname Middle-age, so I get a string of texts to say Dear Emma Jane, we've received your prescription. Emma Jane, we have fulfilled part of your prescription (I don't actually care till it's all there.) They say this is down to my NHS documentation, but my GP and various hospital departments manage to address me as just Emma, so I'm not sure why the pharmacy can't manage it.
I would check with whoever gives the private prescription to make sure they're using Ms. Once that's confirmed/resolved, I'd be talking to the pharmacist again. If they didn't change, I would definitely consider moving (because I can do that without inconveniencing myself, as there's more than one pharmacy close by. I probably don't care enough if I had to travel further I'd change.)
Which reminds me, credit card...