The way I understand your picture... the black lines are roof and the gray lines are posts. And there is no valley between the main barn roof and the roof over the pit.  There is an air gap between the main barn roof and the upper side of the manure roof. I get that.  There should be a rain gutter on the main barn roof to catch rainwater and divert it away.  However, what about the snow that will fall on the main barn roof?  Are you expecting it will melt slow enough that the rain gutters will catch the snow melt and divert it? Is that your expectation?  I am not sure it will work that way.  It seems to me there will still be snow coming off that main barn roof and falling down around the pit.  There needs to be a plan to deal with this. 

I do agree that your design should allow manure pit gases to go up and over the main barn roof. If that main barn roof is a metal roof, will manure gases escaping up and over the main barn roof help to accelerate the deterioration of the main barn roof?  Do the manure pit gases need to be moved away from the main barn more?  Just a thought I had.