I made my first closed end pen by leaving the barrel section longer than the tube by an amount slightly greater than the barrel end hardware. If you are only going to use a converter, then you will need this. I say slightly longer because I initially turned the barrel using a live center in the tail stock and a dead center in the bushing.
Drill the hole in the barrel to a depth equal to the length of the tube plus approx 3/16", but don't drill thru the blank. Drop a tube in the hole, and then a 2nd tube. Mark the end of the 2nd tube and cut it to length. Put some epoxy on the short length and push it all the way to the bottom. Then glue in the standard length brass tube. Turn the barrel to diameter and shape between the bushing and the live center. Partially shape the closed end to length, remove the bushing, and press a dowel into the tube, leaving enough sticking out to put in a Jacobs chuck in the headstock. Carefully finish the closed end, sand the barrel, and finish.
I have only made one, but I like the result. I have some parts and on the next one I plan to eliminate the cap tasse and the clip. Hth.