I've never heard of the 22 Club so can only offer ill informed speculation, which should at least flush out those who are better informed than I. You mention Grapple so perhaps that has a bearing. The MOD itself though wasn't around until the following decade.
'Club' of course brings to mind the RAF Club which opened its doors in 1922, but I don't see why there would be a club named after a club...
Given the British love of whimsy, my guess is that refers to something seemingly inconsequential; the address of a building, a room number within a building, etc. The more illustrious the welcoming letter then the more mundane the origin. That would be my guess, though of course it could simply refer to a Gp or Sqn of that number. Does that bring us back to Grapple?
Wiki says that SAR for the tests at Christmas Island was provided by Whirlwinds of 22 Sqn. Was he on that or involved in any way with them?
https://en.wikipedia.org/wiki/Operation_Grapple
Here is the 22 Sqn Assn site page dealing with the appropriate period,
http://www.22squadronassociation.org.uk/