I googled again with the same results, but I did run across this interesting piece which details NPS policy on caves. There is not a mumbling word concerning camping in caves or rock shelters.
https://pubs.usgs.gov/sir/2008/5023/pdf/01pate.pdf
There is a significant point here. Regulations governing conduct on federal lands depend upon democratically processed laws which in due course lead to explicit regulations. There regs are what the smiling, friendly rangers enforce and not just something they think is a good idea (even if it is a good idea). Been there, done that. As an NPS archaeologist, at one time i held a federal law enforcement commission. A high point in my career ws giving extended testimony is an antiquities case which resulted in what for a very long time was the largest ever fine levied against a perpetrator. So I am glad to see those who plunder sites and otherwise violate valid regs get their just desserts.
But you can't just make assertions about "rules.' They must be grounded in legislation passed by Congress and signed into law. There are limits on the Feds, as there should be. Ungrounded assertions about what is or is not forbidden don't help matters. I suspect that this is where a lot of discussion about law enforcement overreach stems from.
Show me the legislation (actually, don't bother - I am now retired) or show it to any current federal officer and they will perform. Barring that,no deal