After discussing this with @cpinter and @lassoan, there is another way that we could organize the DICOM browser.
The DICOM browser widget could be opened in the central widget like is done in ShapePopulationViewer, ex:
We could still provide the option to use the pop-up window for applications/users that would prefer it.
In the longer term, it would be useful if we could re-organize and simplify the DICOM widget to minimize the modifications that are required in Slicer. Are there any modules that rely on the current structure of the DICOM popup?