Lee dies are pretty hard and will likely tear up anything lesser than solid carbide. Carbide tipped will cut a smaller hole, in hard material, cause the core is soft and compresses some. Also, I've found than trying to ream to size is futile. Even with annealed O-1, I drill, ream and/or using a boring bar to get 0.002 to 0.003" under desired diameter. The die is then finished, in a lathe, with an adjustable lap, with 280 grit clover, or an emery wrapped rod. Final finish is 400, then 600 grit wet/dry. I don't bother measuring, but push through a lubed bullet of the same alloy and hardness I intend to use and measure it with a good 0.0001 res. mic. There is some springback happening, so a 0.381 hole doesn't make for a 0.381 bullet. Go slow and slug and measure often.
My own push-through setup is made from a piece of 7/8-14 threaded rod. The bottom end is bored to accept a 5/8 diameter by 1 1/4 long O-1 bushing. The rest of the length is drilled through to 1/2". A locking nut is positioned and fixed and is tapped on the side for a set screw. The screw passes through the threaded rod and locks in the bushing insert. The bushings are bored to just under size, backbored about 0.010 over, and the entry has a 6 degree lead (12 degree included). Of the 1 1/4" length, there is only about a 3/8" section that is at the sizing diameter. Make finishing to size much simpler and is easier on the bullets. I don't bother hardening the bushings as it just makes finishing harder and usually shrinks and egg's the hole.