Antler sheds don't usually have the "bloodshot" issues common to deer that have died out of hunting season (not necessarily due to poaching ... natural causes and vehicular collisions are possible scenarios).
If you come across one that still has blood in it (the core of the antler is marrow, as it is a bone growth rather than horn, which is a type of modified hair, like hooves are a modified type of fingernail), I would suggest COOKING IT THOROUGHLY to kill off any and all bacteria that may be present in the blood before you attempt to cut it and turn it any further.
They (all types of bone) turn reasonably well with HSS tools, but I prefer carbide, as bone can take the edge off your tools kinda faster than wood. I have found that it is easiest to drill it by hand holding (pliers with leather wrap) on the drill press and then mounting it on a plain mandrel shaft between bushings and taking it to your sander .... hold it at an oblique angle and it will spin the antler section as it wears it away, giving you an easily pre-rounded piece with little fuss (lots of mess but very low chance for any kind of catch or breakage).
If you prefer drilling on the lathe, a similar method can be used there ... hold by hand with leather wrapped pliers, put the back end on the tailstock center and drive it onto a spinning drill bit to the halfway point .... back it out and drill from other side to meet in center. Make sure to push the bit ALL THE WAY THROUGH to ensure you have a straight hole for your brass tube, and glue with epoxy or gorilla glue to fill any internal gaps and seal the interior. You'll still find it much easier to pre-round on a mandrel shaft with a belt sander or disc sander.
Once it's pre-rounded it's a lot easier to keep it in one piece, as the rounded shape wont catch nearly as easily as a rough exterior ... don't forget to seal it in CA whether you choose to use a CA finish or not, helps keep the moisture content in the bone consistent to prevent swelling/shrinkage, which will cause cracks over time.