This is how my setup looks like. Yes, it is very crammed, and yes, this is a bicycle handle just above the tailstock pulley :(
I will be machining a a tiny brass rod down to even a tinier sized rod that will fit into one of the watch hinges. I had no means of figuring out how large of a diameter I needed, except that it was rather small - my caliper is useless for sizes smaller than a few millimeters and I don't have anything else that can measure hole diameters. Fortunatelly, a diode leg just barely fits into the hole, and I can measure the leg's diameter using regular calipers.
One caluable lesson (re)learned here is that very thin things tend to flex and bend a lot. You can see that the diameter of the rod near the end of the cutting tool is correct, but at the very end to the far right it is way too big. Even though the cutting bit is quite sharp, the rod flexes so easily that the bit cuts less and less as it travels further right. Unfortunately, turning between centers doesn't seem to be an option here, so I will just have to finish the work with a needle file with the lathe running and snip off the end I do not need anyway.
After snipping the resulting rod off with a pair of small wirecutters, I need to file the ends to the right size. Yes, this is a needle file in the picture - this thing is quite small.
Now, after everything is done I need to make sure that after the peg is in place, it stays there and does not fall out. I was thinking on just putting that little peg in place, securing the whole watch on an anvil, putting a center punch to the smaller end of the peg and tapping the center punch with a hammer to squish the end making it big enough so it won't fall out of the bracelet. The problem is... the peg is too small to position a center punch on its end. Now, if thought about that ahead of time, when the piece was still in the collet, I could've left a little indentation in that smaller end for the center punch. Since the piece is already out, and is too small to fit into any collet, I have to use the smallest drill bit I have for drilling PCB boards and a milling attachment. I know it's comical looking, but it worked!
Here's how it looks now