I love what you say about separation doesn't teach. I think it does but only to a certain extent. It is one thing to separate as a form of negative reinforcement but you have to teach what appropriate behaviour is and shadow the child to give them opportunities to TRY and learn to behave appropriately.
There are lots of comments about not terminating and I think I was maybe the first and one of very few to say to terminate. I do not terminate lightly and didn't suggest it as such either. The OP was the one who stated she had been dealing with this for 6 months with no improvements and that no discipline helped the child. It doesn't make them a lost cause and it doesn't make them a bully. A bully usually acts with the intension, but how can a child this age purposely antagonize and physically harm another child on such a continuous basis??? I simply made my point in relation to making you think at what point do you not put the well being of the other children ahead of this child who you have clearly devoted so much time and effort to who is to date NOT improving at all.
I think a lot of great suggestions have been made and they could really help, but I urge you to set boundaries and timelines that don't just allow for time to help this child, but also prevent the other children from further negative impact as a result. Separation is for sure going to prevent them from physically harming another child but you can not keep them separate forever and it will teach them how not to behave, if you are lucky, and hopefully break the cycle so that once they are reintroduced into the group again "full time" if you will, they are no longer acting in this way and are able to learn appropriate behaviour. But, I would suggest you set yourself some limits as you have already invested 6 months with no improvement. It is one thing to work with a behavioural issue and see tiny baby steps of progression but its a very different thing if your efforts see no improvement at the cost of either another child being hurt continuously, or you giving so much time to work with the problem child that it takes your attention away from the others more than it should.
I kept a child for nearly 2 years with horrific behavioural issues and he was never physical with another child beyond throwing toys which were never actually thrown at another child but ultimately resulted in some injuries. That child was not a bully, he had areas he needed help in and that's exactly what I dedicated myself to doing for him and the family. It didn't however change the simple fact that his behaviour put the others at risk over an extended period of time and took extra time away from the other children so I could further invest it into helping him. At what point do we say enough is enough. I hate passing a child on for another to deal with, but as with the OP's situation, this is not just about doing what is best for one child, but what is best for the group.
I really hope you can work through this with success. It sounds like you are well invested in this child and I just hope the family realizes how lucky they are.