Yeah, but. . .
Lee is the only manufacturer other than LBT that I am aware of the lathe-turns the mold cavities. It is the exceptional roundness that results that allows these bullets often to beat conventional cherry cut molds for accuracy. Not that cherry cutting can't be done well, too. It is just technically more difficult to maintain exact center and keep chips perfectly cleaned out as the block halves come together. A not uncommon issue with cherry cut molds is bullets that are bigger on the flashing diameter because chips kept the halves from coming together perfectly at the end of the cut. That's not a problem with Lee or LBT.
You want to do a perfect job of removing the base bevel without your bullets winding up out of balance. Also, you want the final bullet dimensions to seal against the sizing. Any way you look at it, it is going to be more expensive to tool up to do this accurately than it is to pay Lee a custom fee to cut new molds without the bevel base for you.
The best approach is to use a milling machine or a lathe, but it can be done on a drill press. You want an X-Y vise or an X-Y table to hold the vice on the drill press platform. You need a center finder to know when you've got the mold cavity axis perfectly coaxial with the spindle of the drill. Otherwise, material will be removed off-center. Drilling soft material without grabbing requires a sharper drill point angle than standard drills have. Also, you want no chatter, so a single flute drill with about a 90° tip angle is best if you are going to drill? A variable radius boring bar chuck instead of a drill will let you turn off a few thousandths at a time until you clean the bevel to the full diameter of the bullet. That is the approach that is least like to introduce an error, but then you need to invest in that special chuck.
You may also get away with stepping up through chucking reamer sizes to final diameter, but I'd only do that if I had the reamers already. And you still need a center finder to get the mold cavity axis coaxial with the drill spindle.
Just fixing the lube error. . .
I can think of other ways to deal with the lube issue. One thing that is likely to help is to buy an RCBS lubricating and sizing die. The RCBS and Lyman dies both fit each other's lubri-sizers. The difference is that the Lyman dies have a vertical string of lube holes on the side, while the RCBS has just one entry hole level.
In the Lyman die, lube hits the bullet at as many holes as line up with the grease reservoir. The RCBS dies are made with a single ground groove like a bullet lube groove on the outside of the die, and the lube flow holes are drilled into that single groove. The outer surface is recessed slightly so the lube can find its way from the reservoir to those holes. That is why lube entry is just at that one level in the die. If you set your bullet up to align the bottom edge of its lube groove with the bottom edge of the RCBS die holes, that's where it will flow in. After filling the bullet lube groove, you have no lube pressure yet at the bevel, so you can raise the handle and push the bullet up past the lube holes before the bevel can fill. The ejector ram has only to rise one hole diameter to block the holes and cut off lube flow. By the time the bullet exits the top, a good length of ejector ram is in the way of the lube flow.
Another thought is to stop using the lubing function of the lubri-sizer at all. Just dip the bullet in mineral spirits as a temporary lube, and run them in and out to size them. After the mineral spirits dry off, use Lee Liquid Alox or White Label Xlox to coat them and let it dry. Works fine with as-cast bullets that will be sized by the gun barrel as well.
I've been shooting a lot of tumble-lubed bullets lately. You know how some people thin Lee Liquid Alox with a little mineral spirits? I thinned some down with melted Minwax Paste Finishing Wax. I have to warm it before I use it, but it dries hard and not sticky, and they don't smoke as much when I shoot them.