In fact, I think scholarly neutrals might be what was wanted on the committee in the first place. Get a couple of Methodists or Presbyterians, a Parsi or Zoroastrian, and a Hindu or Buddhist -- just so long as they're trained historians with a track record of serious evenhanded investigation and skills in textual analysis. No non-believers, though, they have just as much of an axe to grind as Catholics or Jews.
If they do it again - regardless of the makeup of the historians (and I agree they should not be atheist) - I believe it would help to have a preliminary round of negotiations between the Catholic and Jewish sides to lay out the ground rules, including discovery, confidentiality, press relations, etc.