General help

There are probably many ways to do what you want to do.  I will offer one idea.

1) Put the questions subject to review into its own quiz.

2) Have all reviewers take this quiz.

3) When all reviewers finish (i.e., submit with an answer) and all get the questions correct, only these questions are "approved" and eligible to be moved.

4) Have a Moodle administrator move these "approved" questions into question bank X.

Sure, an administrator might like a question but not be able to answer it correctly.  This situation can be handled in a variety of ways, depending on the type of question.  For example, if the question is "multiple choice," the last answer could be "I approve this question."  If it is an Essay question, the administrator can answer with "I approve this question."

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Here's another idea.  In my own situation, I run my own moodle so I don't have to worry about administrators.  But I do import questions banks from publisher textbooks, and I do need to decide which questions I like and do not like.  So, I create a quiz, set the quiz to show answers upon completion, add all the questions to the quiz take the quiz, and then I review every question.  In another browser, I have the question bank open.  I move those questions that I do not like (or vice visa) into a different pool, leaving only my good questions in the pool that I will use for the course.

In your case, you could give all of the administrators the "right" to move questions out of the main question pool that they don't like.  Or they might move these questions into another bank for "Subject to further review" or "Maybe okay."

I do encourage you to figure out an electronic way to do this.  One could use a "paper" approach, but I would go crazy managing all of the paper.  

Also, Moodle is the only major LMS that can help you with this process because of its excellent question bank features.  Moodle's quiz engine is quite powerful, but one needs to spend some time learning it.