I did a full transit turnaround in Taipei a few months ago because of a change in my meeting location. My ticket requires me to fly the HKG-TPE-HKG sectors before continuing on the next sector.

I got my BPs (and I think the lounge passes too) in HKG for both sectors. I did not clear immigration and customs in TPE. When I boarded the flight back, the gate agent asked me as I did not have the exit stamp on my BP. I answered her and she made a note and asked me if there was any checked luggage. I said no, then I boarded the plane.

On my next flight (after the TPE-HKG sector), I was again asked about whether I had any checked luggage in my previous sector when I checked in. I assume besides MCT, they are also concerned about potential luggage problems.

Don't worry about MCT - have a good meal in the TPE lounge. They have pretty good noodles.