Someone published a really good anthology of caving songs in the late 90s /early 2000s. Unfortunately I lent my copy to someone and never saw it again. Contained such classic as "Neoprene Ned the caver". With regard to the BSA I recall this "gem" as a variation to the version above (to be sung to the tune of Clementine), from memory it goes:
Chorus:
Oh me cavers, Oh me cavers, oh me cavers so sublime, out of caverns into taverns where we drink till closing time.
Verse:
See the Beesa up at GG with their sexual machine, Burgess cranking, Simmy w***ing, its the finest sight you've ever seen.
Verse:
See fat Eli stuck in passage, see him wallow in the slime, stick a carbide up his backside, he'll be out in record time.
By t'heck, they don't right them like that anymore