3" throw isn't necessary, as you can tap the lever inside the tank to adjust the throw distance.
I actually kinda mocked this up some time ago, and was shot down by the wife.
Check your local FreeCycle and grab up an old dishwasher (mine was hotpoint) from the early 80's. The drain solenoid has plenty of power to trip the arm when connected to some picture wire and a spring.
As, I'm to lazy to want to say the word "flush" maybe a pressure switch can be incorporated into the seat.
Remember, "If the ladies don't find you handsome, at least they can find you handy."
-Keep your stick on the ice.