Okay, I guess I have to ask. Even granting for the sake of argument that there were no valid Moon missions, why couldn't they just have used the full-sized Saturn V rockets for the secret military payloads?
Why change the size? What practical reason, in the context of your fantasy world, is there to do that? I have to ask because this creates a risk for the military. If they say the rocket is one size, but it's really another size, then someone will probably figure that out and start asking questions. It seems that if you want to keep something secret, it would be a good idea to minimize the ways in which it could be found out. And in my mind that means publishing the actual size of a rocket that everyone can see.