Kohler states that you can use 10W30 in temps that are lower than zero F. You just have to keep a closer eye on the oil level because the engine will use a bit more oil with this lighter grade.
You could also switch to 10W30 in the hydraulic system for the winter and then change it back to 15W40 or 20W50 for the summer.
You should also work on finding and correcting the problems that are evident in the starting system. Begin with the battery. Use a hydrometer to check all six cells. They should show that they have the same specific gravity reading and that the reading is high enough that it indicates a fully charged battery. Then subject it to a load test to find out how well it holds a charge. If the battery cannot pass these tests, replace it.
The second problem is often with the cables that connect to the battery and to the starter/generator. My best advice is to install new ones and make sure that you clean where they connect to thoroughly. The battery should be grounded to the engine block, nowhere else and that grounding spot must be clean and shiny.
The final issue is the start/gen itself. Badly worn brushes, bearings and commutator all contribute to sluggish performance. Either send it out for a rebuild or purchase an exchange starter if you are not capable of doing these repairs.
If you intend to rely on your Case to clear the snow then it is up to you make the Case reliable. You know that you have a problem so ......fix it now.