OK. I was going to wash my hands of the problem, and run a fused wire from the battery + to a lighted rocker switch in the available dash location, to the fuel shut-off solenoid, since a hot wire to the solenoid straight from the Batt+ worked.
Turns out, adding a switch into the circuit, makes it die as well.
In some deductive root cause methodologies it is necessary to turn the problem off/on 5x to prove the root cause is found. This rules out coincidence causing one to go down the wrong solution path. I'm not saying this is your issue, but it does sound like you have conflicting results about a direct wire feed solving all run issues.
It would be easy enough to wire a temporary light to the run solenoid to see if the voltage drops from the circuit momentarily.
Don't rule out my earlier suggestion to remove the wire terminals from the plastic housing to make certain there is unimpeded electrical contact. This holds true for the terminals inside the fuse box on this circuit. Most connections are the male/female tab and it is very easy for there to be insufficient grip between the two metal parts. I've also seen corrosion at the crimp end of the terminal creating issues. It's not like they use automotive grade weather pack connectors.
I don't have the manual for you machine, so it is difficult guessing what transfers from the 855D manual to your machine..
While it seems your focus has been on the power supply side, don't forget to pay close attention to the ground. Wire connections can "look" good, but until you know the metal surfaces are clean and can prove the joint conducts without any losses, it should be assumed bad.
A voltage drop test may prove useful, but his may yield minimal results given the low current draw on this circuit.
It might be worth purchasing the workshop manual to better understand the wiring circuit. They are expensive, but you are putting a lot of effort into the situation and unnecessary parts.