A work permit is separate from an ARC. The ARC gives you residence rights based on your being legally employed, which comes first, assuming you’re not getting residence based on being married to a Taiwanese.
You can always do the application through Ministry #2 and receive the work permit, thereby covering your posterior in terms of legality, and then simply not apply for the ARC using that particular documentation. I don’t imagine it would be a problem to suddenly switch ARC “loyalty” at any point and change your employer of record to Job #2 if you wanted to. Heck, I suddenly changed from a “regular” job to an Article 51 on my ARC and no one said a word. This is not set in stone of course and I do my paperwork in Taipei which is sometimes easier than the other offices, but it seems to me that your best strategy would be to get the legal work permission letter from the Ministry and then lay low. The foreigner who attracts the least attention wins in the end, in my experience.
I’m currently a full-time student with 2 legal Article 51 work permits, but I haven’t changed my ARC to be a student thus far because I think my interests are better served leaving things as they are.
The tax folks shouldn’t have any problems either; I’ve fronted up there for years with so many tax deduction forms that i had to fill out two tax forms to be able to list them all, and no one said a word or even thought about the fact that at the time, a foreigner was supposed to have one (1) job… :shock:
This is not a legal opinion however, just my own experience…