Why should any lounge give you a letter to say you've been denied entry??

It is not the end of the world? It's only a lounge... With some free drinks and snacks. Will it kill you to mingle with the rest of the traveling public in the airport?

BA will not call an airline for you. Your status must show up somewhere and if you can get a letter from BA stating the fact that you are indeed Silver the that's the way to go.