To echo, somewhat, other's comments, an antler pen is more interesting if you have some "bark" showing. If you drill through the center of the antler piece, you are probably not going to get any bark showing on the finished pen. You really need to drill closer to one side to achieve this, assuming your piece is thick enough to do this.
What I do is put the piece between centers, but not centered on the antler. The live and dead centers are offset from the center of the antler piece. As was said, when the piece is spinning on your lathe, you will see the outline (shadow) of the antler, and can see if there is going to be enough space to drill the hole. If not, move it.
You don't need to turn the complete antler piece round before drilling. You just need to get enough turned off so you can put the piece between the "V"s on your vise to hold it straight upright.
Once the piece is mounted in your vise, you can shift the vise around to help you drill closer to an edge with bark.
Also, I find that using a roundnose scraper on this first rounding is smoother than using a gouge. A gouge is more likely to grab, causing the dead center to tear out some of the end of the antler piece. Of course, if you start with a piece that is at least a quarter inch longer than the tube, you are accounting for this possibility.
Hope this helps.
Joe