Paul, you're approach is identical to mine. The key is to make sure that once the barrel is held solidly to the drill press table, either in a vise, or clamped, and 90 deg. to the drill bit, it does not move. This involves leaving enough clearance for the replacement of drill bits, and the tap. Once the hole is drilled, I remove the belt from the drill press, and rotate the tap in the chuck by hand, with lots of Rapid Tap cutting fluid. The taps Chamber's provides are excellent quality, and cut very very nice threads.
As to you're dilemma, Tom, you may have to go to the next larger liner - a 3/8" x 32, and use the recommended tap drill for the final cut before taping the hole.