Fasnachtler,
This is a late reply, and not really an answer to your question. But i think you might find it interresting, i do at least
Right now im reading the autobiografy of the danish tenor player Jesper Thilo (ISBN 87-614-0063-7, in danish i'm afraid), who played a lot with Ben Webster.
Thilo tells that Webster used to call his horn "Old Betsy", but no mention of its brand (but i guess its the BA above mentioned?). Also he tells that Betsy was getting kinda "loose" and therefore he had two horns from Selmer for tryout as replacement (no mentioning of model or year, but i'm guessing somewhat late in Webster carreer?). Ben chose his horn and Thilo bought the other.
At that time Ben gave Betsy's case to Thilo, who later gave it to a jazzmuseum in New Orleans lead by Dan Morgenstern. The case was there reunited with Old Betsy! And i guess you should be able to see both horn and case there.
But Thilo also has another story: when Ben Webster passed away in Holland, a painter there claimed that Ben had promised him his horn. And so he got it.
When the painter died, his wife contacted the Ben Webster Foundation in Copenhagen, because she wanted to get rid of the horn: The horn was playing by itself in the night, haunting the place...
guess this was the "new" horn and not Betsy, but still spooky i'd say!
