The BLM still owns the land and are likely "only" leasing out the "grazing rights"......for the surface and they still own the minerals beneath. I am not 100% on this, but since they own it....they can dictate the terms. Very likely, recreational prospecting is seen as a "non threat" to the grazing rights.
Filing a claim on that land is just only securing your right to further explore and extract minerals from below the surface....and this seems to me that the BLM would allow that as well even though the rancher is leasing only the surface area. The BLM probably has provisions in the lease to the rancher, making it fully understood that the lease does not prevent the BLM from also allowing others to enjoy the use of the multipurpose use lands. I am sure there are penalties provided for those who interfere or obstruct the ranchers rights to graze under his lease.