I'd get in touch with Children's Services and ask for a referral to the Early Intervention Service, I think. They're the experts in this sort of stuff.
What is their behaviour like at school, and are they both performing ok academically? Do they both have a network of friends? Are either of them bullies, or bullied? Do they treat you and other family members with respect? If it's just something that happens between the two of them, rather than part of a wider anger management issue, it will be easier to resolve.
Have you tried teaching her the usual anger management techniques, like walking away, breathing exercises, adopting non-confrontational ways of expressing her anger etc?
In the meantime, I'd set some very strong ground rules, eg no baiting by making personal or abusive comments, no lashing out physically, and sanctions, eg time out for both, in their own rooms, without phones/internet when things get heated, withdrawal of privileges such as clubs, outings hobbies, grounding them etc. And rewards, too - maybe a small gift or treat each if they go a week without incident.
I don't think anyone should make a 16 year old move out, she's still a child, she can't leave FTE and she's still your responsibility.
And what about their father? What does he think, and what input is he prepared to have?
Incidentally, I know a pair of siblings where the younger needled the older to the point where she ended threatening the younger sibling at knifepoint. It was terrifying at the time, but that phase only lasted a year or so and they became really close a couple of years later and still are. This too may pass, and hopefully will.