I would not give in. How do you know that once you cut down those trees that he will be happy and stop his protesting? There might be something else that has a bug up his pants and he will move on to that. You are obviously dealing with someone that feels they are above the law (demanding you do something beyond the town ordinance)
Rent the biggest UHaul you can the day you have an Open House and park it right in front of his house blocking the views to the sign - or does the wacko picket like a protester?
Talk to all of your neighbors. Get everyone together one evening for coffee and to talk. Inform them of this wacko's antics and how it's not just hurting you, but it's hurting the property value of each of their houses. If your house doesn't sell for top dollar, their house will not either when it's time for them to sell. Not to mention, will this wacko have a grudge against them when they go to sell. Who knows, maybe he won't like their license plate on their car or something.

Get them against him.
Call the department that handles this ordinance and act innocent. Tell the person you talk to that you are going to have to break this law because the homeowner next to you is harassing you to cut down more trees than allowed by ruining your ability to sell your house and you were wondering what your fine or punishment will be and give them the details about what this guy is up to. Get a little choked up and sound a little scared and upset. Maybe they will give him a little visit and tell him the fines will be on HIM for blackmailing you like he is.
Go to the courthouse today and file a lawsuit against him. You don't need a lawyer to file suit. Serve him with papers yourself and tell him if the signs come down, the case will be cancelled, if not you will sue him for the difference in price over what you eventually get for the house and what you almost got (make sure you have all the info from the sale that went bad) Don't go small claims, go higher. Let him know he will DEFINITELY be in the hole for over $10K and you will hold him accountable for every single damn penny. If he doesn't stop, then get the lawyer. The thought of a lawsuit may be enough to stop him though.
But whatever you do - do NOT cut down those trees!