First and foremost, you need to check your local laws/ordinances, etc. BEFORE getting chicks and building a coop. You should have known you were short on clearance before you ever decided to start this project.
A little pre-planning here and you could've been pro-active in approaching your neighbors on all adjacent properties and speaking with them about your plans. You would have therefore known from the start that at least one was going to be a problem. Actually, he may not have even been a problem because he may have been more compelled to work with you on it because you took the time to approach him first rather than jumping into the project.
Anyway, assuming that he still would have been a problem, you could have found out whether any of your other neighbors were an issue and whether or not they'd mind if your coop was within 100ft of their property boundaries. If anyone didn't mind, then you could have gone for the variance, kept the coop away from the problem neighbor, and at the very least wouldn't be stuck where you are now.
As for your neighbor's arguments against chickens:
-Chickens will not attract any predators that aren't already there. Skunks, raccoons, opossums, fox, coyotes and dogs all roam many burbs quite freely and very likely some (or all) of them are in your area, too. If they weren't there'd be no chance of them every finding your chickens.
-Chickens will not, in and of themselves, decrease any property values. They may decrease salability of yours/adjacent properties if prospective buyers happened to be opposed to chickens, though. If you don't keep your coop/yard cleaned and odors in check, that could become an issue as well.
-It's your prerogative to request a zoning variance just as it is his to fight it.
-I don't know how amenable the guy is, but maybe a little bit of chicken enlightenment would help, too. Yeah, there are nasty chickens out there, but most are not and you would never let yours on his property or anywhere near him, right?