Ok, lets go over what we know now. It seems you have a bad battery or two right? I assume you are buying a new one now.
If that doesn't fix the problems, lets go back over the basics and define our goals.
The problem is the engine wont crank, correct? Not whether the ignition works or not. Lets just concentrate on getting the engine crankshaft to turn under power of the starter motor.
So after replacing the battery, and the engine still will not crank, lets start over by measuring for voltage on the large bolt on the starter while attempting to start. Do we have a nominal 12 volts on this large bolt? this should be a direct connection from the battery. Now what about the starter solenoid bolt/connection. ? What do we have there? Do we have a good connection from the starter to the frame?
Lets review what the starter solenoid and starter does and how they work. Now I don't know anything about a "mule" but most starters are just a DC motor that draws a large amount of current especially when it is actually spinning the engine over. What is the purpose of the starter solenoid? It is two fold, it is high current "contactor" which is like a relay in that regard, but it also has a mechanical function in that it moves the pinion gear into contact with the ring gear on the flywheel.
How does it doe this? Well it has a coil just like a relay, but this coil is very powerful and has the mechanical strength to move that gear into engagement with the flywheel against a powerful spring and at the same time its electrical function is to make a sizeable set of contacts engage to make the final connection of the battery to the stator of the DC motor portion of the starter.
So this starter solenoid portion of the starter itself draws a sizeable amount of current and is more than the little key switch could handle, so we use a starter relay that can handle this current thru its contacts that is energized by its coil which is connected to the key switch contacts which can only handle a small amount of current.
So, knowing all of this, the question becomes, Does the starter solenoid connection have voltage when the key is turned to the start positon? If not, did the starter relay coil pull in? if not did the key switch apply power to the starter relay coil? Does the other side of the starter relay coil have a ground so that current will flow thru the coil? Are there safety switches?
Answer some of these questions, and let us know what you find.
I realize all of this is just words, but I am a bit far away to be of much actual physical help, and words are all I have to help you with.
