Keep the speed down(if the blank gets hot)...
Remove the drill bit often(clean the flutes of excess material)...
Practice on a few cheap blanks(just to get the feel)...
Both were done with the same basic technique. I turn the blanks round, then part off a small section of the top blank. All the way down to the tube. Then I take them off the lathe and add the cb.
In the first one, I used aluminum and some contrasting resin for the cb. I CA glue everything then clamp. In the second one I used a red plastic and then a piece of the original blank. More subtle. In both cases I turned them to a different shape than the straight slim stick.
This can be done with wood as well. Or mix and match. I often make wood pens with resin CBs.
Are you making it a one piece, or just not using the center band? if you're just removing the center band, the blank shouldn't be THAT much longer. Drill as you normally do.
Is the problem that the drill bit is not long enough? Are you drilling on the lathe? This has worked very well for me: you can first rough turn the blank to fit a collet chuck. This will allow you to drill part way thru, then reverse the blank to drill from the other end. You get the good alignment on the bore doing these steps.