I went to the Stansted 'know your local ATC' tower visit a month ago.
They were very friendly and said , when asked , they weren't 'anti GA' - an accusation often levelled at them. That said it's not the tower who issue clearances, so you'd have to talk to Essex radar.
Essex radar are usually frantically busy and are pretty good judges of a pilot's ability by his R/T manner, so drop your voice a couple of octaves, know exactly what you want to say and say it clearly.
And have an alternate route ready for when you get the almost inevitable 'remain outside controlled airspace'.
In 15 years I've only ever got one transit, not across the runway, but along it from N Weald to our strip east of Lakenheath.
I was initially refused, but when I saw my alternate route to the East was blocked by a wall of CBs I asked again and they slotted me in.
All future attempts were refused so I don't bother now. But, like the man once said, 'nothing ventured, nothing gained'.
Cusco.