I don't have a closed end mandrel that size, but you could probably make a pin chuck version by filing a flat on a short, standard 7mm mandrel rod to accommodate the locking pin. Or perhaps turn it between centers?
The last time I made a desk pen I used a PSI Premium Designer NT Kit. (Premium Designer because I wanted to use a Parker refill instead of a Cross refill and the Premium Designer uses 8mm tubes and a Parker style. It was about 8.5 inches long and I made it by simply using one blank to make the tip end with my standard pen mandrel. For the "clip" end though I drilled 2-inches into one end of a standard 5-inch blank and carefully glued the tube in. I turned it between centers using the bushing and a 60-degree dead center on the headstock and a pointed live center on the tailstock. After turning and finishing it I simply sanded an angle on the free end and then touched up the sanded area with finish. I also turned a matching holder for it.
Good luck with your project, and Congratulations to you and your son!
Dave