SVFR Fixed wing twins do go in and out fairly regularly, normally the AeroMed boys and girls from Oxford.
I am not sure how you know you will only be on the ground for 10 minutes, as your inbound arrival will be subject to the normal delay. As for departing, you will be in the queue with another 20 so the quickest you will do it is 35 mins I reckon!
Fairoaks or Denham as said. MUCH quicker believe me!