I would advise you get your insurance sorted with the BCA as I know Plym do quite a bit of caving.
But it might not be as grave as it sounds, I believe you're known as Plymouth adventure and Expo. As such your code of conduct says that all members must have BMC membership. The code of conduct goes on to reiterate much of what people have said ie get BCA insurance for people who need it.
Due to the fact that all your members must have BMC membership, I would assume that they are partly covered under the BMC policy.
"The members of the club resident in the UK are insured for liability claims made by third parties (including other club members) for any club activity and as an individual for climbing, mountaineering and abseiling and all associated activities such as mountain walking and scrambling. Additionally activities such as ski mountaineering, occasional mountain biking and caving, are covered provided the activities are non-competitive and are SECONDARY to the activities of the club (i.e. are not the main activity of the club - it would be assumed that a club whose prime activity is caving, for example, would be affiliated to and have insurance through the British Caving Association (BCA)). The cover does NOT apply to those who are acting as professional mountain guides, instructors, etc."
In the short term you might be alright, but I would ensure you are following your code of conduct. As this is the ultimate guide on the clubs responsibilities to its members.