Re: Weak hive and strong hive - time for a nuc?
This is my opinion. I would do a combination of both. Move the "machine queen" to the nuc with a frame of capped brood and place it where the existing hive is now. Put in four frames of foundation or three foundation and one drawn comb and you have a flyback split. Take a frame with swarm cells and put it in the weak hive along with another frame of capped brood. Finally, squish all but two of the remaining swarm cells in the original hive and move it to another stand at least 5 feet away from the original location. These manipulations should stop the swarm, strengthen the weak hive, and give you a third hive with the good queen. Be prepared to hive that nuc within a few weeks as it should really take off. Be sure to feed the nuc and the weak hive, the parent hive should be fine with their reserved stores. Use an internal feeder and maybe put robbing screens on the nuc and weak hive.
When you introduced the new queen, you did locate and dispatch the old one, right? If not, you will need to find her and put her in a cage or give her an alcohol bath.
Thankfully, the bees are smarter than I am. They are doing well, in spite of my efforts to help them.