Original User: mikem
No, there is no way to do this in a hosted manner. You could download the application, modify the code, and host it on your own web server. This would require some coding.
I know that the Basic Template references the group that contains basemaps. You can edit the index.html file and change the group to your group with custom basemaps. (Note that your basemaps groups consists of web maps. Only the basemap inside the web map will be the actual custom basemap. Layers on top of the basemap won't display as part of the custom basemap.)
Thanks,
Mike