Actually, I think it's a good idea from the standpoint of the Batfamily to keep Hush around. A fake Bruce Wayne will keep people from wondering why both Bruce and Batman sort of disappeared around the same time, and from our point of view will keep Bruce's spot warm, figuratively, for his eventual arrival. And since they know he's a fake, it should be easy enough to control him for their own sake.
Of course, it would be far easier to get a friendly shapeshifter to act as Bruce Wayne for a while, but I can see why they might not just give Hush the boot right away.