You have not conclusively proven that the ignition module is bad. That is a lot of work to replace a part that may not be bad.
It may be as simple as low cranking speed. The small batteries used in these come in a variety of flavors from crap to pretty stout. I just replaced the battery in my 1872 due to cranking too slow to fire in the cold.
If you need a battery, the only number to look at is CCA (Cold Cranking Amps, measured at 0*F). The other number is just CA (Cranking Amps, measured at 32*F); no need to pay attention to both numbers.
Only look at the CCA number. There is no way to make that number high without making a better battery. Some crap batteries will not list the CCA; if they won't show it, you can bet it is low.
The best I found was at Advance Auto Parts, 340 CCA ("Lawn & Garden Pro"). The worst I found was also at Advance, 160 CCA. The price difference was substantial for a reason, it costs more to make a better battery.
With the 340 CCA battery, the cranking speed is substantially better. Also make sure your cables are in good shape and use dielectric grease on all connections to keep the corrosion away.
You can't troubleshoot it until it doesn't work again. The next time it won't fire, try some starting fluid. Often they are cranking too slow to fire on gasoline but will fire on ether. If it fires on ether but won't stay running on gas, you have a fuel delivery problem. If it fires on ether, stays running on gas, and never has a problem at any other time, it is probably just cranking too slow.
If it won't fire on ether and the cranking speed is reasonable, then it may be the ignition module. Check for spark immediately during this troubleshooting session before something changes, not the next day when conditions are different.
Get two spare plugs for checking for spark. The Kohler manual recommends using one of these two plugs gapped at 0.130" when you are checking for spark external to the engine to simulate combustion chamber conditions (compression). This is generally done by just removing a portion of the ground electrode of the test plug to create the large gap. (Measure to be sure of the gap.) Connect both test plugs to the plug wires (one with normal gap and one with the large gap) and ground the plug threads using large electric test clips wired to the engine ground. (Don't remove a plug from the engine or you will increase the cranking speed due to the lowered compression resistance, and you will be changing the conditions the engine is experiencing.) The Kohler manual does a good job of explaining this and has a diagram. The Kohler service manual is available here:
http://www.kohlerengines.com/onlinec.../tp_2204_b.pdf
The Kohler manual has a second test using an ohmmeter that can give assurance of a module problem. If you are not getting spark, make sure you don't have the kill wire grounding accidentally. Unhooking it at the engine for the test is one way to troubleshoot this.
Ignition modules (magnetos) can be intermittent, but it won't stay that predictable. If it is really a problem, it will become more and more apparent.
Good troubleshooting is proving something is good or bad before you move on. Guessing and just replacing parts will make you crazy and poor! Good luck!