Jake,
The header widget in the launch pad theme controls what is displayed based on available screen space inside the Widget.js resize function. It reserves 250px on the right side of the screen for the onscreen widget icons and then does many more calculations to determine what it can display based on available space. You can adjust this function if necessary (I would start with changing the offsetRight) or just change some configuration settings like remove the search widget if not needed, change the default theme layout from default 1 to style 2, that places the onscreen widgets to the right of the header vertically instead of horizontally, remove links from the header if not necessary.
The thing to consider is that when viewing the app in WAB you have the builder app taking up a lot of screen space on the left so check the app by clicking launch so that you get the full screen real estate for testing.
