I hete to tell you this because I know you are anxious to spend $200 for another tool; but a collet and pin chuck are conveniences, not a requirement.
You can do it all on a mandrel with the tailstock center replacing the knurled nut to hold everything together. The only difference is that you can't turn the end of the pen, and it will have to be trimmed and finished as the last step.
The advantage is that you can turn the entire pen (both barrels) at the same time.
Go to my website and look at the Long Line pen. That is a closed-end pen. The only difference is the length of the end.
There is also a detailed presentation of setting up the mandrel, turning, and finishing both the Long Line and a closed-end pen on my "Advanced Pencrafting" video.