If your GSD population does get out of control, here is an on-point article (abstract) for you.

A flood introduced Gizzard Shad into Montgomery State Fishing Lake in 2007. They did thrive in that lake, so they decided to attempt to kill them in 2013. (That lake is 105 acres and 30 feet deep. Located two counties east of you.)

They did draw down the water and treat with rotenone as canyoncreek suggested.

Eliminating Gizzard Shad in a Small Kansas Reservoir

The treatment did get most of the Gizzard Shad, but the long-term effects on the LMB/BG were difficult to quantify. (So many other variables in the real world.)