Basemap Gallery transparency setting disappear if user switches basemaps even if they switch back. I have the imagery basemap with a transparency to mute it but it does not preserve transparency once different basemap is selected.
Unfortunately, this is as designed. A custom basemap can have a transparency set for a Web Map and by design once the basemap is switched the transparency is reset. The transparency set for a basemap will be persistent and saved with the Web Map as long as the basemap is not switched. These rules for transparency are dictated by ArcGIS Online and the Web Map therefore, carry over through to the Web AppBuilder application. Two different Web Maps would need to be created, one fore each basemap so the set transparency will persist.