We would suggest that you get in touch with a good youth organization or an organization like our Hunters Against Hunger to come to the property and do guided hunts.
I would think that if you made your wishes known about the consequence of taking a button buck, you may well do some good. This is one case where baiting may be an answer to locating doe' s and fawns. Baiting overwhelmingly brings in doe' s and fawns. You could put minimal bait in a shooting path and sit in a blind. If you have the amount of doe' s you say, bring the animals in for culling in a controlled situation.
Once the limit has been reached under your control you stop the hunt... Just a thought... Drastic situations require drastic thought...
Also, with a population of doe' s that large it should be easy to document crop damage. I would talk with the farmers and try to start an influence in the direction of a well balanced herd. The farmers could rent their land out for $$$ if they have the large bucks. I would think that if the farmers started making a noise about crop damage and start a suit against the managing authority for reparations for crop damage minds could be changed. Farmers need to be shown how $$$ is lost and could be made passively and they' ll come around.