It might take a few minutes to hook up, but I would make up a switch, or use your bump starter and hook one of the leads to the main side of the starter solenoid itself and the other to the start position (S terminal) of that same starter solenoid. When your issues flair up, grab that bump starter and see it everything works as normal. Your effectively eliminating the electrical system out of the equation. If that works, wire up a 'hot start' old style Ford solenoid remotely and hook it up to your existing starter. The original start wire will trigger your new solenoid. Run 12 gauge wire to (From the Positive Battery) and from the new solenoid down to your old starter solenoid S terminal.. We did tons of these when I worked as a fleet mechanic @ Ralston Purina.. Old systems degrade and it's just as easy to wire up this set up than go through all of agony..
EDIT. Here is a kit
https://www.holley.com/products/elec...s/parts/51340G
https://documents.holley.com/199r11281.pdf
Continuous Duty one ..
https://pitstopusa.com/i-22954444-ta...-and-ford.html