It's not a code problem, really. With the default setup (let's assume my project is called "Game"), the process is
1) "Game-Desktop" program with void main executes "new LwjglApplication(Game(),otherArgs)"
2) "Game-Core" "Game.java" program executes. This class implements "ApplicationListener" (and any attempt to remove this implementation messes up the args in LwjglApplication).
So, that means basically the first major part of the entire application to execute is the piece that renders the screen. This is totally backwards from what I'm used to. As posted above, I really enjoy the entry point to be a "main" class that executes the separate pieces of the application (Back end/calculations, Screen, Input).
What I ended up doing is modifying the "Game-Desktop" program to do just this. It may be completely against convention and I will need to duplicate these steps for any other platforms, but it allows me to set the game up as I like. I do not like the screen rendering program to be responsible for performing backend calculations.