Won't crank but starts when starter solenoid shorted
 

1999 Savana 3500. (This is an Itasca motorhome)
Intermittently, the engine won't crank when key is positioned in start. Sometimes it starts fine but most of the time I have to short the starter solenoid with a screwdriver to get the starter to engage. When the key is in Start and it does not engage the starter, I hear the coach battery solenoid trip to add the power from those batteries to the starter circuit but no starter engagement so I can assume the ignition switch is OK. All batteries are good.
I have changed the starter and starter solenoid, starter relay and bypassed the neutral safety switch. Any ideas where to go next? Thank you.

I would look at the battery cables for condition, tight connections and corrosion at both ends not just at the battery.

Leaving aside the backup system engaging, have you
put a meter to the primary solenoid coil to see if it is
seeing what the ignition key's saying? Path to the other
battery may take a different route. Or maybe the solenoid
is just bad. Have you tried putting B+ to the solenoid
coil to see if the solenoid works?

Yes. What voltage are you getting from the ignition key.....at the solenoid? Voltage drop could cause the click, but not enough to engage the solenoid. Does the screwdriver trick work on the new starter as well?
My dad is having this exact problem with a tractor of his......hard to find.
I am guessing bad ground on the starter side......or bad wiring on the starting circuit.....between the key and the solenoid.....