Luck= critical hits, not rare drops.
Beat Time is a universal timing system, not anything to do with rare drops.
If you look at: http://tychopso.psupedia.org/
you can have a look at dropcharts, and see the droprates of rare items for yourself. It's possible you're looking in areas with a lot of very difficult drops, which would explain why you're not finding much.
If you just want to see a red box drop, run Ult Fake in Yellow a few times.
Bezmor: There's actually 2 1/205 rare monster drops, the other being Purplenum Mil Lilies, for a Psycho Wand.