rules?
if the big hive is diseased you may be spreading it. ?? workaholic maybe

other than that , "equalizing" is often done , bees, brood and honey. 1 or 2 or all.
keep moving the frames.
Do not rule out a 2 queen senario, maybe had a supersedure and 2 sisters did not fight to death and you have a 2 queen thing going.
sure you can move frames , I do that often, whole supers can be done.
Tipping over would not be good, so sure take some frames away.
Add a few frames to any of your other hives.
when does your flow end? As the end nears, they should start capping some/most of it.
The hive swap would mitigate the "swarm" issue as some of the field bees would be on a different hive.
if the hive makes the winter, be a good candidate for splits.
Be advised the bigger they are the more likely they get Mites. So end of july pull all the supers and treat if you do, can shuffle the supers around treat 1/2 with the supers on then other 1/2 then switch.
GG