I think I see what you are doing. Basically your overflow would dump into your skimmer intake, and your skimmer sits on your sump and dumps into that. Right?
This really depends on your skimmer. The vast majority of skimmers work in a 150-300gph range and most overflow systems crank a lot more water than that which is a bad thing for the skimmer.
Provided you don't exceed the GPH
rating of the skimmer this should work fine. However, I'm not sure what type of skimmer you have or how you'd inject air into it unless it was airstone driven or an injection type like a Remora.