So, first placed I would have thought it was on the "attributes" tab......maybe hacked as an added proxy rule? Just guessing.
Have you tried the "click to add a new basemap" in the properties for the Basemap Gallery widget?
If you can figure out the item ID from AGOL you would expect it to see the key you have entered there. If not, this seems this is an overlooked feature "we" should ask for. Enter it on the Esri Arcgis Ideas | Ideas Submission Portal page, and I'll vote for it.
But my guess is it may need to be a custom/modified widget. Suggestions, since this is related to AGOL too, it might help to move this discussion to the ArcGIS Online or maybe Robert Scheitlin, GISP 's Web AppBuilder Custom Widgets page. That may help it get more views from others doing the same sort of thing. And Robert plans to work on custom JS widgets as he did for Flex. Good location to join/follow.
Edit: forgot to add the image.