I'd check intto the larger tractors and machines that folks use for bigger jobs than mowing the lawn. i.e. lawn/garden tractors. Those bigger machines must have just what you need. Or, build a framework to hold several push buttons-one on each end, and a spring to center the handle when you let go of it. A vertical piece fastened to the tractor, with a cross piece near the top end, and the ends turned in 90 degrees, with the push buttons in those holes, wired to raise or lower the weight. When you would pull the handle back to the rear button, the lift would raise, and when you push the handle forward, it would push the button in to lower the piece. Or maybe the weight would pull it back down! Some experimenting with this idea would/might solve your problem.
Also, those larger tractor sales folks might give some advice, too. You can always go there and peer at how they are connected. by: Rustyj

The rocker switch on your Simplicity is most likely controlling a relay(s), but you need to understand how the switch and circuit you have works. It should be easy to reverse engineer the wiring (if you understand DC electrics) and substitute a more glove friendly switch. You might go looking for a switch that just has a longer stalk you can mount on the steering column below the steering wheel.

There are many two position momentary contact switches for cars that have bat handles instead of rockers so a simple change of the switch you have might be very easy.

You only need a two axis electrical switch and most joysticks on heavier duty GTs and Subcuts are hydraulic controls for FELs.