Even if you do get your bubble tip anemone to split on its own or you cut it yourself it will still grow back to the size that it is right now within a very short period of time. So aside from maybe making a few bucks selling the clone its not going to solve your problem.
Here are two pictures of a tank in town that used to have a lot of problems about five years ago. The bubble tip anemones stayed small and divided like crazy. There may have been as many as 20 around this time. Shortly after they all bleached and died.
Here are two more recent pictures from that same tank with two new bubble tip anemones. They no longer stay small and multiply, these two were added separately about three years ago after the problems with the tank were fixed. Its possible that one of them may have been related to the originals as some of the original clones were sold locally and this one did come from a local tank. But the other one was purchased from a retailer out of town.
I have a few theories but nothing concrete. On one hand it looks like they only divide when something is really bothering them but on the other it could just be that it depends on the individual specimen.