put one queen in the nuc, kill the queen in the strong hive, go back in a while and do a walk away split on that hive. that way none of your queens carry the genetics of the mean queen unless they are passed down through her drones.
Doing a walk away split will result in queens raise from brood that was laid by the unwanted queen.
Here's a youtube video of a guy requeening an aggressive hive. You'll have to let us know if your bees are as nasty as these. I've seen videos of Africanized honey bees that weren't as aggressive as these.
This shows that the bees do remain extremely defensive even after the removal of the queen. Since you already have your queens, I would go through and kill the old queen and requeen that hive. Then it depends on how soon you want to do a split. You can queen the nuc, wait two weeks then check the hive for new eggs. By then, any eggs laid by the old queen will be capped, so you'll be looking for eggs and young larva.
As soon as you have a couple frames with eggs and larva, you shouldn't have a problem doing a split with fresh stock. Good luckRequeening A Vicious Honeybee Hive
After you dispatch the mean queen, don't just toss her away. Get a small bottle with a little bit of alcohol and drop her in and mash her up. You can then use it as a swarm bait lure. Stick a Qtip in and then rub the inside of the swarm trap with it and drop the Qtip into the trap.