Sounds good, Gang - thank you.
I suppose I assumed there was something special that I needed to do to make sure the wood stays on the shaft, like a special adapter or something.
I considered a bolt, but there would be threads on the end and using it in a drill press would crush the threads over time. Eventually this would affect the way the burnisher turned. An Old Drill bit is a pretty solid idea too.
I will buy a boring metal rod and cut it to size, score the metal so the epoxy has some place to grab and give it a go.
In retrospect, I should have done this FIRST as now I have a hole drilled and will need to re-turn the burnisher to make sure it's true. Will be easier to turn these moving forward using the post as a handle in the lathe chuck.
Thanks for the advice and the SPEED of the advice. I posted late last night and this morning I have the direction I need.