Hello, did you cure this fault??
I have the same symptoms on a machine I've been asked to look at..........

Just replaced the starter. When the customer first called on this the complaint was the starter was grinding. Sure enough when I got there the started was not engaging into the flywheel. Then it started doing what I described in the post. I removed the starter and the drive teeth were damaged. Put a new starter on it. Connected the DVOM and checked the battery voltage. The starter worked 3 or 4 times and then started acting up. The battery voltage never dropped below 10.7 volts when the starter worked. As I explained before, I can't get a voltage reading on the solenoid wire or a drop reading because the starter won't stay engaged long enough. I will throw another battery in it on Tues. just to verify, but I believe the problem is elsewhere.

yup
get a load tester and test that load drop on the battery
it may read good with a vom but could be failing under load.
if the ecm or vcm sees the voltage drop too much it will disengage the starter.
you could always pull the starter and directly hook it to a known good battery just to verify that it works correctly but i'd do that load test first ;o)

Is the battery good and fully charged? Are the cable connections good?