Well, just got off the phone with ESRI Tech Support. They have logged it a bug:
The Bug reference number is [#BUG-000091206 - When using Web AppBuilder for ArcGIS (Developers Edition) 1.2, the widgets stacked together in a group in the Header Controller do not show the scrollbar whenever required.]
I'll keep tabs on this and let folks know when it is resolved or what the potential resolution is...