I use the 3 Beall buffing wheels that mount in the lathe and get a decent shine.
I start sanding with 320 or 400 at 1800rpm or less, around and then lengthwise with the lathe stopped. Might use 600 if I feel like it.
#0000 steel wool, lathe on and then lathe off lengthwise.
Wipe off all the abrasive.
Tripoli (brown) at about 1200 to 1500 rpm. Around the barrel, then diagonally both angles and finally lengthwise. Clean off all buffing compound with soft cloth or paper towel including the ends.
White diamond same rpm the same way. Around, angle both directions and lengthwise. Clean off all compound.
Carnauba wax same directions as above.
Caswell have some better compounds and wheels worth substituting or adding to the sequence. You'll have to figure out which ones.
Note if you see scratches you can go down without cleaning off the compound but you must clean before going up to a finer one. If you don't clean you take coarse compound up and contaminate that wheel.
The reason I buff at slower speed than most is it keeps the plastic cool and acts like a softer wheel.
This is a Black Japanese Ebonite pen I made years ago following the above routine.
https://www.penturners.org/threads/little-black-number-custom.103377/ Wow hard to believe that was almost a dozen years ago.
Pete
Looks like from my old thread I used lower RPMs but that doesn't stop you from playing around to see what works the best for you.