I ended up leaving the fixed cup in place.

After cleaning everything, I re-greased and re-assembled. Cleaned up the cranks, and put them back in place. It spins quite smoothly (and quietly) now.

I figure than the placebo effect from knowing that I have overhauled the bottom bracket will be worth at least another 1-2 MPH on the flats.

Thank you for all of the advice. I am now on to cleaning up the front and rear deraileurs (but I will save that for another thread).