Well, my 0.02p...
Every time I've asked for a transit from Stansted/Southampton I've gotten it!
Although the caveat is that I've only ever asked twice from either (still 100%!)
.
In every case, my requested routing was directly across the runway (i.e. not flying into the final approach track, but rather crossing the runway at 90 deg straight through).
Standard Noise - always found Bristol very helpful & accomodating, many thanks.