Again, I think you might have contacted the international clinic there (seems to be called “Dream Way International Healthcare Center”, judging from an e-mail in English I sent to the hospital last week to see whether the blood testing service was open on Friday - I presume that’s where the English-speaking staff is so that’s who answered).
I wouldn’t do that. I don’t think you actually need a psychiatrist, do you? My (limited) understanding is that ADHD would also be within the remit of neurology, the practitioners of which I believe are more inclined to medicate than deal with the underlying behavioral/psychological aspects.
This is what I would do (in order):
(i) Go ask in the pharmacy I linked to above, just to confirm that you can’t simply buy the medicine OTC. You can also try a couple of others if you want, taking the packaging of your current meds and the certificate.
(ii) Make an appointment to see a neurologist at Taipei Medical University Hospital. You can do this online and then just arrive there 30 mins before to complete the registration process in the main foyer area (they have volunteers to help). Bring your passport and documents from the previous doctor. Don’t say anything about being an international patient, except for answering “meiyou” every time someone asks for your NHI card. The basic cost for this appointment should be on the order of NT$500. Explain the situation and ask the doctor whether he/she can prescribe what you want. If they say no or that you need to see a psychiatrist for that, NT$500ish will be your total cost.
(iii) Repeat this process with one of the neurologists at Mackay (the English website is less convenient and you won’t be able to find out much info about each doctor, so just choose one at random - I’ve seen maybe three there and they could all speak passable English and gave me the medication I asked for). Again, if this doesn’t work your total cost will be about NT$500 and you can report back for further suggestions (possible other options would be finding a non-hospital clinic/family doctor to give you a repeat prescription, but I don’t know much about that).
