A hinged roof makes all the difference here. Extra points if you figure out how to easily prop it open when working inside.Thank,you also for the advice my plan,right now is,to put the mesh across the top also the run will be 8 feet long I plan on hinging the top for easier access. That is design 1 design 2 is make it 6 feet tall and throw bird needing over the top, but not sure design two will keep the predators out.
4 ft high is plenty for the comfort of the birds, even 3 ft is enough for the run, they aren't roosting out there.
Biggest oversight with predators is not expecting them to dig under the perimeter. I assume you thought of that too.