Thanks for the suggestions. In particular, suggestions 1 and 2 are most appealing since they require no tinkering.
I had been doing suggestion 1, but I thought I would more carefully observe TBX behavior when I do it. I noticed that upon clicking on the Name in the Text Panel, a highlight appears to reveal a bounding box that seems dedicated to display $Name. The box is taller than is needed for one line of text (perhaps that depends on the font size?), providing ample space to wrap $Name to display more content.
I offer it as a suggestion to provide an option to wrap/resize text to fit $Name into the header of the Text Panel. The space is already there, and code for dynamic text-wrapping is already a part of TBX in other places (e.g., displaying note titles in Map view). It seems like an implementable feature. Consider it a small feature suggestion.