You have found a bug. The problem is the coordinate system of the WMS service is not being preserved when embedded and is getting switched to geographic coordinates. Thus, your shapefiles don't overlay properly.
The only work around I can think of (I believe this will work, but don't have your shapefiles so I can't test it) is to recreate your map keeping the default basemap in web mercator. Then add your WMS and shapefiles over the top of this. The WMS service does support Web Mercator, so it will overlay. You can then set the transparency of the basemap to 100% so that it is not visible.
Thanks,
Mike