Hi Brian,
Heavy is subjective. Many fancy canes are made from ebony, which is also a dense heavy wood, and they are not considered heavy. As a matter of fact, I like a cane with a little heft to it. The physics involved with using a cane is a little different - the weight bearing is along the length of the shaft, not against it. In other words canes are for leaning on, not hittin people with it. Therefore the shaft really doesn't have to be that big around, but it does help to have the grain running lengthwise rather than at any oblique angle. Really, if you keep your shaft under an inch, it should result in a manageable weight, particularly if you taper it to the tip.
Canes in three parts usually have the handle as one part, and the shaft in two parts. That's going to have you turning pieces that are from ten to fourteen inches long, depending on where you have the joint. You should be OK on the mini lathe if you have a bed extension. If you are planning on a three piece shaft, plus a handle, then you won't even be close to needing a bed extension.
I hope this helps! I would really like to see pictures of this cane when you're done.
Scott.