What condition is the lead screw in? If its worn or damaged, then you have an imediate disadvantage, you will not have enough pulling power, the screw may well stop drawing the wing tips into the wood, especially in beech which isnt a fibrous wood, and simply spin freely in its own cone shaped hole, getting you nowhere fast. Assuming the bit IS in good condition and as Ozzy said, IS sharp, then I would sudgest using an extra long tommy bar and have yourself and an assistant to share the work. I can manage 1 1/4 inch OK with a standard brace, although I over did it a few years ago drilling something like 120 3 inch deep ones in 2 days for mortices, ended up wit tennis elbow
.....but 2 inch is a "bit" much (sorry
) The good thing with the long augers is they are far easier to sight up for the correct angles, harder to drift off as it is with a brace and bit. By the way what sweep does your brace have? I have one somehwere with a 7 inch sweep (ie creates a 14 inch circle as its used) gives more power torque than a standard one)
Of course your other option would be to chop out square tapered mortices instead, that only needs a small pilot hole....just a thought
good luck with your project