I could be wrong but I looked this up recently, and this is what I remember:
BMC insurance only covers caving on an organized trip of that club (where the main purpose of that club is not caving). In other words there is no time that BMC insurance can cover a person going caving with a BCA caving club (unless that club is also a BMC club and it is a club trip).
BMC member going caving with a BMC club which has organized a caving trip as a one-off/occasional special trip: covered (for that trip).
BMC member going caving with a non-BMC club: not covered.
BMC member going caving independently: not covered.
BMC member going climbing independently: covered (this is the key difference).
SamT said:
So their BMC insurance covers them 3rd party underground (occasionally).
As far as you (as a non-BMC club) are concerned, no it doesn't.
http://www.bmc-insurance-centre.co.uk/
Note that no mention of 'caving' is made for the 'members' insurance (because members are NOT insured for caving), and that (as already stated in this thread) the club insurance covers it only as a secondary activity during a club meet only.