Our local cemetery has numerous large holes 3" duameter, some with substantial dirt mounds. Saw the critter running today--16-18 inches at least, with furry tail 6-8 inches long. Holes sure look like major underground burrows.
Clackamas County Oregon
Many of the characteristics you describe are right on or close to dimensions and habits of California ground squirrels - sometimes known as Douglas ground squirrels, diggers, or gray diggers. Their burrow entrances can range 3"-6" wide. Here is a link to an old, but still informative brochure on control methods and ecology http://icwdm.org/handbook/rodents/ro_b151.pdf
Because the pesticides registered for use on ground squirrels are Restricted Use, only licensed applicators can buy and apply them. Especially given the sensitive nature of the space they're using and the damage they're causing, your cemetery board/city/or county should probably consult with and get an estimate from a licensed private contractor (Wildlife Control Operator) or their own licensed staff professional to devise a population reduction and control plan that will cause the least disruption to the cemetery itself and the visiting family members. A list of current Wildlife Control Operators can be found here http://www.dfw.state.or.us/wildlife/license_permits_apps/wildlife_control_operator_contacts.asp