Well I guess this is one benefit of the new ID cards with one letter.
That’s what foreigners in Japan do. When asked for ID they don’t show the ARC but instead use a drivers license and health card as ID, that way service providers have no idea if you are a citizen.
I wrote to the Museum and received an email response from the Director apologizing and saying that APRC holders are included in promotional deals and that they need to do a better job of ‘educating’ their staff about the differences between various kinds of foreign IDs (that I assume means the difference between APRC and ARC).
Notice that the National Palace Museum opening hours are from 09:00-17:00 (Closed on Mondays) So you only have 30 minutes to visit the exhibitions in the Main Building (國人憑證件及持中華民國外僑永久居留證)