By all means, get the game official involved, if for no other reason than to remove the " NO HUNTING " signs from public Lands. Don't argue the issue with the guy yourself. There is no point to doing so. He thinks he is right; you think you are right.
A GPS and a good map should resolve the situation. You can get a County map for every county at the local Highway Department office. If there is no such county office, then ask at the County Clerk's office.
I use them hunting all the time, along with plat books which are often available at the local farm bureau, or local banks. You don't even have to have the whole book: Just photocopy the few pages that involve the lands where you intend to hunt.Often the Plat Book is more up to date than the County maps, and the plat books will give you an idea of the names of owners. If you want to be sure about a particular owner for a property, ask the Supervisor of Assessments, because his office prepares the tax bills annually, and gets all the names of new owners any time property is sold, or inherited. A phone call to the named owner will verify he still owns the property, and whether he has given any control over the property to the pushy neighbor.
As a habit, I get the name and phone number of any landowner who has given me permission to hunt on his property, and write that down on the back of my maps, and then have the owner draw the lines indicating his property boundaries. That way, if some neighbor calls the Game officials on me complaining that I am trespassing on his land, I can show the officer my map, the boundaries, and who owns the land, with his phone number to contact. That gives the game official ammunition to go back after the complainant. The officers particularly like to see plat books, becuase they often have printed right on the tract of land the name of the owner. Because plat books have larger pictures of smaller areas, its far easier for an officer to figure out where he is, and whose land he is standing on. That is important to settle these kinds of disputes.
Other than arresting poachers, I don't know much of anything that makes game officials happier than taking down No Trespassing and No Hunting signs that have been illegally placed on public ground.