I got excited about this and felt stupid for not trying this myself.
Then I looked at my splined BB tool, branded Nashbar - sadly it doesn't have any wrench flats.
I always thought the design was rather silly, putting the ratchet so far outboard of the small engagement of the splines leads to a lot of force and torque that doesn't work towards the goal of tightening or loosening the BB cups. That and the drive socket is larger than normal, such that I have a wrench that I use only on that socket. I left a perfectly good Shimano BB with the plastic cup inside a trashed frame just because I couldn't get it out. I guess I'll just pony up for a better splined tool with wrench flats.