JN,
Firstly, how nice to have a civilised exchange! I agree that Sqn designation and ownership seems like small beer - and it may be, for all I know - but if I were to re-phrase your question as:
"What is the seniority of the individual commanding this organisation, who is it subordinated to for C2 purposes, and where do the personnel of this unit live and work"
Which is not a million miles from what you've proposed, then perhaps you'll agree to the sensitivity. And 51 was for a long time not as open as you suggest.
Finally, in a former life, I've been on the answering end of FOIA requests (although not at the MoD), and it's not a big deal. It'll get you the official answer and may well be more than you expect. It's only 30 days, and may be quicker - it should also be more illuminating than the
PR line.
Cheers,
S41