Don,
For that deep-hole drilling in the buttstock, I would insist on using either 1. engine lathe or 2. drill press. Both with suitable large vise blocks & clamps. An unshaped blank is much easier to clamp.
You need good tool rigidity and perfect alignment to do the job right. Only one chance, you know. You do NOT want to experience that gut-wrenching sensation of seeing the drill bit come out the side of a beautiful block of AA Select Fancy grade of walnut!
A speed-bore spade bit, properly sharpened will do the trick. Use a sturdy extension to make it long enough to reach thru in one pass. An auger-type bit would work, but you would have to grind off the square shank to fit the chuck.
1. Lathe is easier. Use spacers and clamps to hold the work to the carriage/crossfeed/compound. If you have a milling attachment, use it. Use the carriage handwheel to feed the work into the tool bit held in a three-jaw chuck. The carriage will give you plenty of lateral feed to do it in one pass. Use the tailstock handwheel for more pressure if needed.
2. On the drillpress, use vise and clamps to hold it securely. The drill press quil may or may not have enough movement to drill through in one pass. If you don't have enough movement, then stop, adjust the table up (with drill inside hole) and continue.
Alignment is critical. Measure and clamp everything carefully.
Use a speed suitable for that size bit (1/2", 5/8", 3/4", whatever). Maybe 400-1000 rpm. You will be able to "feel" the proper feed rate.
HTH
John