That's a interesting tool. I found this, they sell screws as small as 0000-160.
A division of Swissturn/USA, Inc., the J.I. Morris Miniature Fasteners maintains inventory in a wide range of precision-machined miniature fasteners for immediate delivery. Quality and reliability guaranteed.
jimorrisco.com
There is also a PDF catalog. I feel like the watchmaker in the old joke - "What did the watchmaker say when he saw a 1/4 - 20 tap for the first time? I didn't know they made 'em that big"! I'm looking at the 0000-160 screws and threaded rod they sell thinking - I didn't know they sell 'em that small. I know about the small thread series, but I never seen a supplier offer anything smaller than 0-80.
Many years ago, I used a 0-80 tap and die to make a low temperature differential Stirling engine from plans in James R. Senft's book. I used my smallest T wrench with the handle removed and my thumb and index finger on the knurl of the T wrench to tap the threads. I did it while seated and resting my arms on the bench. I got lucky and did not break the tap. I felt like an hour at the time, but actually it didn't take much longer than tapping a 4-40 thread. The act of tapping a 0-80 thread is the same as a larger tap, just be very, very gentle I kept telling myself. Very nerve racking the first time. Afterwards, I'm like "0-80 tap? No big deal."
Based on what I've seen of your posts, you will have
zero problems tapping a 2mm thread.
Thanks for sharing the J.I. Morris tool photos.