Duncan,
Let me help you avoid a bunch of work, and trouble.
Here's what I went through to narrow this similar problem down.
Changed the starter 4 times, figuring I was getting bad rebuilds.
Cleaned the ground to the trans, and added a grade 6 ground wire off the
frame to the bottom starter bolt.
Replaced the neutral safety switch.
Replaced the ignition switch.
And it still didn't want to start! (occasionally.)
I was frustrated --big time!
So I tried this---
I slide the passenger seat all the way forward, then took the screws out
of the shifter console and lifted it up. Then I took an alligator
clipped jumper wire, and attached it to the positive terminal, and
touched it to the solenoid signal wire on the neutral safety switch-----
It started every time.
I seems there is a voltage drop in the starting circuit sometimes, and
it prevents the starter from engaging --sometimes.
Now, someone on the list suggested to add a starter relay, and that
sounded like a real good idea, but I was so pissed off at this
cantankerous van of mine, I was  at the end of my rope with patience.
So in a fit of rage, I resolved the problem simply.
I took a 8 gauge wire, added a crimp connector to one end of it, so I
could bolt it to the end of the battery terminal clamp.
Drilled a hole in the right hand seat base, and added push button
starter switch, ( off a caterpillar D-8) then under the rug I ran the
wire to the starter signal wire, soldering it about 3 inches below the
neutral safety switch.(elapsed time about 1 hour)
Guess what?
If and when my Frankenstein monster doesn't want to start----I hit the
button and ---Bang---- it starts---every time.
No swearing, no cursing, no banging on the dash, no slamming doors, just
a grin on my face that it hasen't got the best of me. (yet)
Next time it doesn't want to go, give the jumper wire to the battery a
try---it might help narrow your problem down.

Good Luck,

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Terry 85 GL