I don't think I can post the obituary due to copyright etc, but he died in 1936, not 1937. He was buried at Harehills Cemetery with full military honours on 13 Feb 1936, section Q 365. His headstone is a little worn but reads as follows:
RSM J Keane
Royal Signals
In memory of a very cheerful comrade
who died 9th February 1936
erected by all ranks and friends
of the 49th (W.R) divisional signals T.A
and Air Contingent Signals S.R
As far as I'm aware his records should still be with the MoD. For info, I have a family member who recently served full-time with the Reserves (which is something I didn't know could happen, until it did!) so its possible he was still in full-time service, even though he was with the territorials.
ADDED: Although his headstone says only J Keane, his burial record is as Arthur John Keane.