talk to your laywer

First of all, your laywer should have more experience with this. I was panicked first and felt much better after talking to my laywer.

Secondly, it is sometimes hard/impossible to fight back each of their silly questions directly. In the exampel you listed, if you can't argue against that, you should find other evidences showing them you are really exceptional! e.g., strong letters even from the same people who already wrote you the first ones.