So, after 2 days of trying to figure this out I found some sort of solution for this to at least get past this. So I have a simple ear with this layout:
This structure was exploded out like this in my "deploy_export" folder. I was hoping ShrinkWrap would take this structure and create the ear from it. However, it seems that for some reason, the sub-archives are not deployed correctly by creating the deployment this way. So, what I did was removed the war and the jar from the ear structure and put them into their own exploded directory. I then executed this deployment logic instead which makes calls to add the war and jar as a module. This deployed correctly.
EnterpriseArchive theEar = ShrinkWrap
WebArchive theWar = ShrinkWrap
JavaArchive theJar = ShrinkWrap