Actually, Pete only joined The Beatles around this time, just prior to the first trip to Hamburg, so I assume that this letter was sent out just in case Pete didn't accept the Beatles' invitation to join them.
According to Lewisohn, the drummer-less Beatles first saw Pete Best playing with the Blackjacks at the Casbah on 6th August and he had an audition with them on 12th August. The Beatles and new drummer Pete left for Hamburg on 16th August.
So, according to the BBC story, this letter was sent by Paul on the same day they auditioned Pete to join the band, so it must have been a back-up plan in case Pete decided not to go to Hamburg.
I wonder who the drummer was who had advertised for a band in the "Echo"? I bet he kicked himself when he realised he almost had the chance to join The Beatles but missed out on it!!