Two things you would have to do to make that work.

1: Make sure you don't drop the fish with the weights into water that's too deep. Like water that's below the thermocline where the oxygen is depleted in the summer time.

2: They would have to be able to release the fish after a while when they get the fish down to the proper depth. Something that would release the fish by dissolving after some short time in the water. Once the fish is acclimated back to the depth they would then be able to go where they want.

This all would be a lot of trouble. It might be expensive but if you really want to save the fish caught during a tournament then it might be better to have a weigh in more frequently or to have a judge in each boat to weight and meaure the fish and then take a pictures before releasing the fish right away.
Fish that are left in the live well after being caught in very deep water > 33 ft deep would still suffer a lot of stress if you held them in the live well all day long.

One thing I did was add a long PVC pipe to the intake out my live well. I weighted one end of the pipe and connected the other end of the pipe to my live well intake hole with some clear tubing that's flexable. I wrapped some tefflon tape around the tubing to make it fit the hole better. I eventually lost the pipe but it worked how I wanted for a while. I could draw water up from the depths into my livewell. That water was 10 to 15 deg F cooler than the surface waters.

But I was fishing in a small lake that only allowed electric trolling motors and not gas engines at that time. I would have lost the pipe much sooner if I were fishing in a lake where I could have used my gas motor. I would have forgot that the pipe was back there and taken off before pulling the pipe out of the water. It would have fallen right off the boat the first time I fired up the motor and took off at full speed.

Perhaps if one could design a pipe that would stay attached when moving by swiveling upwards and then it would drop back down and hang vertically when you slow down.

The key was to draw cooler water up from the 10 ft deeps water. Even five or six feet down in early summer will give you some cooler water for the live well. Cooler or colder water holds more oxygen. Stressed fish need a lot of dissolved oxygen to help them recover from the stress of being caught, stress of having their slime rubbed off and the stress of being kept in the confined space of a live well with the other fish.

If we didn't have so many people fishing for bass these days it would not be necessary to worry about the bass population. There would be a lot more bass in the lake when the fishing pressure is reduced. I grew up in that environment and fished for a week or two at a time when there was very little other boats on the huge lake I was fishing. KY lake.