Generally speaking it's recommended to divide/move spring blooming plants in the fall (still giving them time to settle in, so November might be a bit late - at least it would be over here :) and summer and fall bloomers in the spring. However, lily of the valley is such a tough customer that I'd move it any time except in blazing hot sun. Just get your axe out and get to work :) C.

Rhizome is root mass below the surface. What you did was to cut a shoot that was just starting and will now die. In order to get a viable start of a clumping bamboo, you will have to get a decent size chunk of rhizome and, preferably, at least one fully developed culm. Unfortunately, clumpers have dense, woody rhizome systems. The last time I divided one, it took 1/2 hour with me pounding on the foot rest of a sharp, steel spade with a sledge to force it through the roots like a chisel. Some clumpers are less difficult...but you get the idea. You're not just going to be able to take a small piece and call it good. Unless you can get viable root mass, you're not going to get anywhere.

If they are Stella's you could divide some now, depending upon size. It's better after bloom. I dig the whole clump then divide, but for older ones, I just shovel them directly out of the ground.

Division is best done in the spring and early fall, at a time that will give the plants a few weeks to adjust before the weather becomes extremely hot or cold. For fall dividing, at least 6 weeks before the first frost date is a general rule. Those with severe winters (especially zones 3 & 4)are at greatest risk of losing plants over the winter, so spring is a better time to divide. In hot climates (for example zones 9 & 10) more plants are lost in the summer months due to heat stress, so dividing in the fall may be best.