Daniel Corwin May: When faced with a situation like yours, I think it best to back up and conduct a formal trouble-shooting exercise. For instance, you seem to have replaced just about everything electrical on the tractor, but have you been able to verify that the problem is indeed electrical?
Formal Trouble-shooting proceeds by trying to "cut the problem in half" and working on the half that seems to be the problem and then cutting it in half and so on until the problem is identified and resolved.
In the case of an engine, three things are needed for it to run: Fire, Fuel, and Compression. You've been working hard at eliminating Fire as the issue, have you considered Fuel? Valve issues would come under the Compression head.
Consider the possible symptoms of fuel starvation.
If the engine is starved for fuel, it might die at full throttle, but start again as soon as the bowl refills. It might also run at half-throttle, but refuse to run at full-throttle because it can't get the volume of fuel it needs from the tank.
I would work my way back up from the carburetor to the tank, replacing the line and fuel filter (if you use one) and flushing out the tank really well, including the removal of the "tap" to which the fuel line attaches --it often has a screen attached to it on the inside of the tank.
It doesn't cost much to replace the fuel line and filter and this way you've done a lot to eliminate the "third leg" of the engine-running triangle. Another thing to watch is the type of fuel filter you're using, I don't think the 1450 uses a fuel pump, so you need a filter designed for gravity-fed systems. Kohler makes such a filter and I would use it before using one designed for automotive applications.
If re-working the fuel line and tank doesn't resolve the issue, I would invest in a diagnostic tool to verify that the the engine is indeed losing its fire. Advance Auto sells 2-3 different types of spark testers, NAPA sells one made by Briggs & Stratton, the Kohler manual recommends taking the electrode off a known-good plug: anything to get you to where you can see if you have a good spark. In your case, a tool that you could hook up in-line with the plug and use while the engine is running would help you verify that when the engine dies it is doing so because it has no spark, or "Fire."
Or, you could start with the "Fire" test and if it is not conclusive, move to the "Fuel" issue. Only if you have verified that you have Fire and Fuel, would I start investigating the Compression issue. In your case, I imagine, but have no experience of, a sticking valve with a weak spring. It could possibly be sticking while hot, and not needing very much cooling before the spring pops it back into place; but I think the Fuel issue is far more likely if you haven't yet eliminated it as an issue. Also, you can typically hear a valve "pop" if it is re-setting itself while cooling.
My two cents. I know that when your tractor doesn't run right, your patience can be severely tested; but it is experiences like this from which "experts" are born.
