No, other than the incredibly organic method of removing them by hand, I don't know of a non-chemical method for eliminating mealy bugs. I'd like to think there's a bug that eats them, but I don't know one, so that's just a pot of wishful thinking.

There's a spineless variety of O. fragilis? The native one we have here is immensely spineful.

Are these small cuttings? Rubbing alcohol might work....it takes repeated treatments to get the hatchlings. Also a strong jet of water from a garden hose will knock the critters off the larger plant. Again, you have to do this repeatedly and really get into the nooks and crannies....only organic methods I can think to try!

There are so many of these bugs and they move pretty quicklyThere is O . fragiles v. denuda. There are several varieties that are called Cat's paw cactus: potato Cactus,nevada cushion, Black Cat, and Alberta sunset. They have them at Coldhardcactus.
It wasn't the cocineal bug it was a coreid bug that is causing me problems.. Sorry
http://entomology.ifas.ufl.edu/creatures/beneficial/c_v_aequoris.htm