To answer the original question.............when I flew from Stapleford it was usual to do a downwind join at 1200'. The safest place to cross would be across the middle at 2200' because other aircraft joining would be quite a bit lower over the top of the airfield whatever the runway and direction of approach.
I suggest that it would be good practise to call and let them know, thereby letting other people know at the same time. Instructors particularly are used listening out for that sort of message and hopefully anyone else crossing LAM would be listening out as well.
I don't think that the possibility of hearing a gruff response should put you off - it can't be as gruff now as it was when I was there!