The red stag I have worked with is indeed very porous inside, more so than whitetail, but not as much as elk. Try to work with the tines out toward the tips, where the diameter at the small end is not much bigger than the bushing OD of your pen kit and you might be able to turn it with the large end still in the solid areas. If not, when you first hit the marrow, stop the lathe and pour on some thin ca, let dry and work slowly.
In order to drill as much through the center of the blank, I mount the undrilled section between centers and look down at it while it's spinning. You will be able to see the shadow or residual image of the final diameter of the piece USING THOSE CENTERPOINTS. You can adjust the centerpoints to maximize the size of this solid cylinder, and that will center the blank. Then turn roughly round so you can easily mount in your drilling vise or scroll chuck to drill. This process makes it much more likely that you will not break through the side when drilling on a curved section, and allows you to use smaller sections that are more solid.
OH, it's also a good idea to hit both ends well with CA before mounting on centers for turning, as this will help keep the centers eating too deeply into the marrow and possibly splitting the blank