What about inducing a shad kill by supercooling the water? Gizzard shad die at ~ 40F normally. Diffusers placed in the deepest water of the lake and run during some cold weather could certainly drop the water temps enough to kill them.
You wouldn't get rid of all the shad, but it would be a good starting point to knock them back while you establish a better predator population to control them in the future.
DO levels would be a concern when all those fish start to decompose though....
