When I make pens from biro nibs with ink tubes I simply drill a 3mm hole the length of the blank for the tube, enlarge the hole one end to suite the nib & then turn the blank between centres to the size I need using callipers to achieve the required size.
This one is laburnum
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I can’t see how you‘d achieve this on a mandrel certainly not in compression as the components would be too small, you could try adapting your lathe so any mandrel was held in tension but I still can’t see that being successful.
Personally although you require a fair number I would drill your blanks & turn them between centres, at between 1-3” long they are pretty small & a simple jig could be made to hold against the turning to ensure it’s the correct finished size.
good luck.