I don't think you're missing a trick but is the cup metal or nylon? Either should simply pull off by hand, no twisting required. If it's nylon and the old Campag/FAG style that takes a six point external tool then, with all due respect, fit it to a trebuchet and go and buy something proper instead. You might get it in with no drama but you sure as hell won't be able to get it out after a few thousand miles . . .
I'm assuming the LH cup has been correctly identified (it's the RH thread) and that this BB isn't a clone of an early 90s Shimano unit that had the fixed fitting on the non-drive side and a nylon cup inserted chainset side.
ETA Of course the LH cup has no flange if it's a UNxx Shimano stylee . . .
