Some Registry references.
Bagpit Fissures Barrington, Nicholas and Stanton, William 1970 The complete caves of Mendip p.22 Cheddar : Barton Productions with Cheddar Valley Press, 131pp, illus, maps
Bagpit Fissures Barrington, Nicholas and Stanton, William 1972 The complete caves of Mendip p.26 Cheddar : Barton Productions with Cheddar Valley Press, 2nd ed., 155pp, illus, maps
Bagpit Fissures Barrington, Nicholas and Stanton, William 1977 Mendip : the complete caves and a view of the hills. p.29 Cheddar : Barton Productions with Cheddar Valley Press, 236pp, illus, maps
Bagpit Fissures Witcombe, Richard G 2008 Who Was Aveline Anyway? [Derivation of the cave name p. 12] [2nd. Ed] 212pp.illus. WCC
http://www.mcra.org.uk/registry/sitedetails.php?id=710
As you say no footpath but the track from Cheddar Head Farm runs close. The site is very close to The Southwest Overthust and the line of closed karst basins associated with it.