What you want to do is something that, for myself, I wonder what I would do if I had to do it over. I already had lots of tools for regular (flatwork) woodworking. But Would I start over with lathe drilling and get a good chuck and drill chuck, . . . or would I get a DP first? Knowing what I know now, . . . That is a tough one.
To me, your choice will boil down to: Do you want to turn pens out fairly quick with simple blanks? ( go for the Drill Press); Or do you think you will be more interested in going for the perfection of pen making, specifically in segments and designs? If this, then lathe chuck will be the choice.
One last tip. When a habit is formed and method is adopted, most people don't like changing, unless they are not getting satisfactory results. Even then, human nature will dictate that changes will stay within the basic method familiar with. (Example, DP method not satisfactory: spend more money and get a different blank vise; Lathe chuck not quite right? - spend money and get the best lathe chuck you can afford and will use for the rest of your life.)
If one is getting satisfactory results, it is hard to change over to a totally different method even if a little better performance is promised. Human Nature. What I am saying is that "most" people usually will stay with the same method and seek to improve within. So choose what you think you want to do for the long run and go for it based on that.
But BOTH tools are valuable. Can't do without either in my setup.