Hello, the reason why the error is being received when attempting to create a data package from the Publisher toolbar, is because the Bing Basemap is a map service hosted by esri/microsoft that is being viewed as read-only data in ArcMap. Because the data is located on external servers, there is nothing that the data package can package. Therefore that layer/basemap is invalid.
However, the basemap does get written to the .pmf that can be sent to another user. Providing that they have internet access that basemap should display automatically when the .pmf gets opened in ArcReader.
I hope this helps.
Thank you.