What make and model bus and engine do you have? Sounds like a pre 90 MCI split dual radiator system...
Interesting, I've never seen anything like it. With 4 fans all blowing into the fan box, the only way to exhaust is to have the door open or a screen door, as you said. If you reversed the flow of air thru the original split system radiators, could you get enough CFM to suck out the hot air thru the fan box and out the radiators?...
I've added some pictures. The system is pretty sophisticated, as you can tell they put a lot of thought and engineering into it. There was apparently a huge scoop on the back that the original designer built that shot air into the fan compartment. The previous owner removed it, and just drove it with the door off. He tossed the scoop. The system works, but I think it can be improved. I am surrounded by mountain passes for the most part. I'm not afraid of completely redoing it, modifying it, etc. For some reason I am only able to get one picture to attach, but I think it tells the story.