Now that Peter Lane's book THE CATENIAN ASSOCIATION 1908-1983: A MICROCOSM OF THE DEVELOPMENT OF THE CATHOLIC MIDDLE CLASS  has arrived and I've had a chance to skim it, I find that there are three references to Tolkien within:
(1) page 137: ". . . there was the call by Grand Presidents and Provincial Councils for Circles to be on their guard, when enrolling new men, to ensure that men of the right 'quality' were enrolled rather than men in 'quantity'. Certainly the Association enrolled some distinguished men in this dire period [1923-1939]: there were academics such as Bodkin of Birmingham, Tolkien of Oxford, Phillimore of Edinburgh and Dixon of North Lancs, which at a completely different level there were th Test cricketers Andy Sandham of Croydon and 'Patsy' (christened Elias) Hendren of West London, who entertained many a Circle with their cricketing stories."
(2) page 153: "So, although wartime difficulties led to some Circles . . . being deprived of their Charters [through lack of unevacuated members], those very difficulties in the shape of evacuation led to expansion elsewhere. And another Circle which owed its origin, in part, to evacuation was Oxford, where many Colleges were taken over by various government departments. The opening of the Oxford Circle in October 1944 was notable, at least with hindsight, for the initiation of Frank Pakenham (later Lord Longford) and Professor J.R.R. Tolkien, the Founder Vice-President. In the light of the current interest in Tolkien's work, one would have wished for a recording of the speeches at the second annual dinner of the Oxford Circle, February 1945, when Tolkien proposed the toast to Provincial Council 'in a most amusing way which included an actual toast in Anglo-Saxon'. He was a member of the Association until 1956."
(3) page 160: "In October 1950, the Brothers of the Oxford Circle congratulated their former President, J.R.R. Tolkien, on the publication of THE HOBBIT, but could not have known that they were witnesses to the beginnings of a cult."
--As for the Catenian Association itself, it was (and is) a fraternal order like the Lions Club, Rotary, Optimists, Kiwanis, et al., founded in Manchester in 1908. Members originally referred to each other as "Chums", which to me sounds rather Babbitty and which I find rather hard to picture in Tolkien's case; I'm not sure if the habit persisted into the days of his membership. They were particularly interested in getting Catholic schools started, since there were far too few of these to meet the demand early in the century.
The book does cast light on just how few Catholics there were in England at the time -- while the group reached Leeds as early as 1910 and Birmingham in 1912, before World War II there were too few middle-class Catholic businessmen in Oxford to achive critical mass