I have the same mower. It was given to me, had IIRC 21 hours on it (70 now). When I got it, the engine was trashed. I got it started but it wasn't any count. She ran it out of oil. VERY common with Kohler engines of the same family. There is a drain plug on the RH side of the engine, toward the front. Square head pipe plug. They get loose and oil leaks out, and people don't check it, it runs outta oil and bingo, engine failure. Also common are valve cover oil leaks on engines that have sheet metal valve covers. The plastic ones don't leak much if at all. And lastly, the old style ignition coils and starters are high failure items on the Courage V-twins. Kohler has updated all of those and the new stuff is much more robust. All except that stupid drain plug!
That said, you can get a new engine (complete) from Kubota, but they're expensive. They are COMPLETE, you bolt it on and hook up your cable, fuel line, and wirings. It will also come with the updated starter (has a 40% larger shaft plug about 50% more torque as I remember).
Second option-short block. Kubota doesn't sell a shortblock. Kohler does but you will have to do a lot of research to find it. I got one for mine, was $667 at the time. Probably $1200+ now. The bad part is, you will have to remove the engine and take the heads off for inspection before you order a short block. You will be reusing the heads, intake, gov linkage, choke/throttle plate assembly, engine electrical, flywheel, blower housings, and all of the exhaust. If the heads are damaged, you'll probably be better to replace the engine assembly. The part number I used was Kohler 32 522 06
Or last option. Find yourself a kawasaki engine and bolt it on. I think the muffler from the Kohler will fit. The biggest issue will be the throttle cable(s) and then possibly the wiring harness. I want one on mine and had 3 of them to "try" but they all went in the scrap dumpster before I could nab one.