It doesn't really matter, because you're drilling to an internal cavity nobody's gonna see. The trick here is going to be getting something to behave itself on the going-in side, and nothing is really designed for that. That's why the pocket jig exists.
Thing is, you need a much larger hole to start with than any bit outside of an auger or paddle will get you, and neither will get you where you want to go by any stretch of the imagination. Plus, pocket jigs don't usually run to the size you'd need. So, it has to be routed first, then drilled after the fact. I don't know of a commercially available router jig that will make that cavity, so you'd probably have to come up with something custom. It's not impossible, but it's not going to be easy and you're unlikely to have many or any practice shots.
Fortunately, there's a lotta unused meat in the body at that point, so coming up with a jig for two dimensions shouldn't be too tough. Once that's done, then you can use a pocket jig to tunnel through to the control cavity with a twist drill, brad point, Forstner or whatever. Shouldn't need to be a big hole, so a twister will probably do.
