Sounds like another mechanic might be in order too. Assuming the starter is a 6 volt, i'd have them both load tested. If the battery flunks get a new 1000 CCA ( 'bout $85 'round here) take it home and put a slow trickle charger on it till fully charged. Most batteries come of the shelf at about 75 - 80% charge.
We used to charge em before installing them back in the olden days..This will add a bunch of life to an otherwise good battery.
The starter, as you described it, is in better shape then mine and mine works.
You might try jumping it at the solenoid. If it cranks good then check out the ign. sw and the associated wiring to and from it.
Won't hold and idle....worn out throttle shafts leaking air??