I have always struggled with "B to E" or "B to B" or "B to C" and the way these phrases are bandied about almost aggressively by people wanting to show how businesslike and in the know they are. For a long time I had no idea what they were on about!*
I am currently writing a chapter for a book on what I have called staff advocacy. Cut out the middle men, let your staff talk directly online with your customers. They are anyway but have to pretend they are not or do so with only partial information. Why not accept this is happening and make a virtue of it. Imagine - real conversations between real staff and real customers!
I reckon E to C is the way to go ....
*They refer to communication between Business to Employee, Business to Business, and Business to Customer for those as unenlightened as I was