Is there a way to push apps to iPads OTA without entering a ID/password?

I work for a public school system that has about 1500 iPads. Currently we have one school with about 600 iPads and students are using them in a 1:1 capacity. Each classroom has a cart with a laptop that controls the iPads through Apple Configurator. Each cart has its own apple id.

We just purchased a Mac Mini Server with the newest server version on it. I set up profile manager and was really excited about the ability to remotely manage restrictions and apps. But then I discover that when trying to use the app that's pushed through PM, you need to enter in the Apple ID and Password from the Apple ID I used to get the push certificate for the server. Does anyone know a workflow that makes it easiest to get Apps pushed through Profile Manager and work on the iPads?

Are there any MDMs out there that can actually push Apps out without every iPad signing into the Apple ID? I just called Apple and the guy didn't know anything that would help me.