Your issue is not the App Builder program it is your use and understanding of how to add operational layers. If you look in my attached graphic you will see when I choose a map service and chick on a map service (represented by the folder icon, number one in the image) and then click the Include button. It will add the Whole map service as a dynamic layer (that supports the drawing of label that are setup in ArcMap). You you double click the folder then you get a list of individual layers of that map service and when you choose one of those layers and click include that LAYER of the Map Service gets add and is added to the map as a feature layer (which does not support labeling). So just click the map service folder (not choosing individual layers) and you will be able to add dynamic layers to the map.
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Hey Matt, I'd recommend just go with editing the .xml. The App. Builder is really for people that are not comfortable with editing the .xml, and given that you're a GIS Coordinator and GISP, you should have no issue with that. Your life will be much better just getting the job done in the .xml vs. using a GUI.