I have the same problem- Kohler SV715 courage 22 hp that I've had for 5 years. First issue ever. Runs great and then after about 10-15 minutes starts running rough with occasional backfire especially under heavy load. Mice had chewed wiring to cylinder #2 spark plug/ignition module. Replaced wiring harness and ignition modules on both cylinders. Also replaced fuel filter and pump. Engine really came to life and could mow for a while, but now back to original problem. Discovered sloinoid was not clicking when turning key to first position. Rebuilt carburetor and solinoid works. However, problem remains. I can get it to run, albeit erratic, by playing with the throttle- moving it into choke mode for a brief time and then back to running mode. One thing to note, I did not realize that cylinder 2 was messed up because of mouse issue- so basically cylinder #1 was carrying the load.
Any feedback is greatly appreciated. Thanks...