Over 183 days is for every single (tax) year! Foreigners can be Taiwan Tax Residents only half of the time.
This is because Taiwan Tax office thinks foreigners can not have domicile in Taiwan unlike other Taiwanese government institutions. They say the only permanent proof for residence is a valid HHR which only Taiwanese citizens can have (APRC is not accepted).
Have a read here:
The bank might give you trouble if you try to change your tax residency:
Persistence (and wasting a lot of your time) might work though.