I’m sorry to hear they’ve given up, especially since the issue is specific to that server.
The debug output looks like just the code for the WordPress admin bar, so it doesn’t offer much in the way of clues.
I have two suggestions:
1. Check the apache and PHP logs around the time of an error, they are critical for solving this kind of problem.
2. Run some tests with your other plugins deactivated and using a standard theme. It’s possible the issue is caused by some other code.
Finally, if you can find some sequence of steps to reproduce the failure that would be a big help – intermittent issues are always tough. If not the logs (#1) will help.