Welcome from Colorado!
Sounds like you have a great lathe to turn pens on. Size of the lathe doesn't matter...you should be able to turn pens on anything from a mini up through a full-sized 24" swing/42" long lathe.
Instead of buying a smaller lathe, put that money towards a chuck, some appropriate jaws (pen vise jaws ideally), and either a pen mandrel...or IMO better: turn between centers. For that, you just need a
60 degree dead center and a
60 degree live center, then just buy
"TBC" bushings for you pens, rather than the normal bushings. It is possible to turn on a mandrel, but there are many complications that can arise there. There are some very high end, high quality mandrels that can eliminate most/all of these issues, but you will pay a good penny for one. TBC is generally what most skilled turners here recommend, and it is not that expensive to get going (you may already have a 60 degree live center with the lathe, so you might only need a 60 degree dead center, which shouldn't cost much more than $10-15 or so.)
I started turning pens "for real" the beginning of summer last year. I'd done some slimlines the year before, but I didn't REALLY get into pens until about 8-9 months ago. I started with a basic mandrel that came with a Rockler slimline starter kit. The shaft had a curve to it, and that always seemed to cause some problems. Bought a higher end mandrel from PSI, and that, too, had a slight curve...and it didn't seem to fit the point of my live center as well, and seemed to have even worse problems at the tailstock end (could never get my pens to turn down to the bushing properly, due to vibrations). I then bought the PSI TBC "mandrel" set, which is a dead and live center with a small mandrel shaft on each. This eliminates the shaft going all the way through the pen blank, but the live center does not seem to have a quality bearing, and I still had some problems with the tailstock end bushing vibrating, which limited how closely I could turn (and my turned blanks were either out of round or non-concentric...two of the more common problems with pen blank turning).
I moved to doing just TBC on the recommendation of many pen makers here. Best thing I've done for the craft. Turning directly between centers eliminates almost all of the vibration issues I've had before. I originally started just putting the bushings I had for the pen kits I was making between centers. After a few months, a small groove started appearing on both my dead and live centers. I have started doing two things. Either drilling with a 60 degree chamfering bit a chamfered edge on all of my normal bushings, so that instead of a hard 90 degree angle biting into the dead & live centers, there is a 60 degree surface that simply "sits" on the centers. Alternatively, I'm starting to buy, from the tbcbushings.com site I linked above, the appropriate TBC bushings for the pens I am turning. These TBC bushings are already designed with a 60 degree chamfer in them, and they are also designed with a more solid extension that goes into the brass pen tube. The 60 degree chamfered edge on the bushings is very important when doing TBC pen turning.
Anyway...TBC with the right bushing design...and you should avoid all the potential problems with mandrels (unless you opt to just get one of the high end mandrels right out of the gate, which is also an option). Some people have the philosophy of starting out with crappier tools, learning from mistake, and eventually progressing to better tools. I used to think that way, but having done things both ways a number of times now, I'm rather staunchly of the philosophy that you should just buy the right tools for the job, and avoid all the hassles.
Its more pleasant, and you should be able to more quickly get good quality results, and do so more consistently. The only thing you can really learn from using the wrong tools, letter tools, or insufficient tools, is that you should have started with the right tools in the first place!!
