Several things come to mind about that. Splitting the piece first would then force me to use a chuck (which I need one anyways) which, for the smaller lathes like ours, are only three jaws which would be uneven pressure on one half of the piece. I do like the idea of turning it with a chuck versus head and tail stock centers since that would allow me to taper the bottom more.

Using tape to hold two halves together would also create a gap in the middle that would certainly catch the on the tools risking chip out versus a smooth cut/scrape. For wood that might work okay but not for the acrylics.

Since I would have to split it anyways, it would, I think, make more sense to turn it then split it.

Like I said earlier, we're still trying to figure this all out so my son can maybe add this to his pen making business.

Keep the ideas coming!