Interesting question.
Unfortunately, I can only speculate since my experience is limited. My speculation: If you want the average of the Coefficient of Variation, I do not see any reason why the mean would not work. Normally, CV is used to compare one process with another. If these are running over a period of time, the mean of CV should be an acceptable measure. I would recommend that you ask this question to whoever said the square root of sum of squares is required. Frankly, I do not see it, but I could be missing something.
Regards,
Don
------------------
Just the ramblings of an Old Wizard Warrior.