Nice practical test, Ben. Never having seen a Boys up close and in person it makes sense to me.
If the mystery square pin adapter was fitted to the rifle, it would appear to place the line of the barrel at, or just above the top of the two side wing nuts that clamp everything into place. That is very likely just high enough for the force of the recoil to rock the upper portion of the mount back against the springs, before they hit the angled stops and absorb a chunk of the recoil energy. The springs then return the rifle to the 'ready position' as it were.
For balance, I think the springs would need to be arranged in some fashion between the gun itself and the mount body. These springs look contained within the mount itself.
Ever considered emailing some photos of the mount to the Enfield Pattern Room in England? They might just have details on the missing piece.
David
PS: The contact for the Enfield Pattern Room is
enquiries@armouries.org.uk
The photos of the mount alone, in situ in the carrier and Tony's two of the potential adapter for the Boys rifle would probably narrow the search down for them, Ben.