I used to work for a company that had the aircraft mechanics that were union and we were exempt management which caused some problems. We had a pilot that had a grievance filed against him for pulling an airplane out of the hangar thereby performing "union work".
Aside from a good old fashioned house cleaning I'd say your problems will be difficult to solve. It's going to have to start with a department meeting to clear the air followed by some good will gestures by both sides. Perhaps start sending some of the mechanics on a few 4+ day trips might help.
Seems like every time I've seen departmental problems like this there is always 1 guy/gal that's the problem. If that's the case, find out who's stirring the pot and show 'em the door. Sounds bad, but I haven't seen things effectively solved any other way and if there's nails being put in tires it's definately reached that point. No HR department on the planet would tolerate that type behavior for a minute. If everyone watches this idiot carrying out all of his/her stuff in a cardboard box I'll bet things fall right in line. A department has to be one big happy family. It doesn't take many people to make everyone's life miserable, as you're obviously finding out.
I've heard of flight departments being closed over this type of discord. If word of this reaches senior management you all could feel the wrath. Can you imagine the CEO's response when he hears the department in charge of flying him/her is engaging in this type of behavior? It won't be pretty.
I can't resist recounting a classic example of dealing with a department "headache".... This the first Director of Aviation I worked for. Classic.
Director: So I hear that you've been messing around with "pilot A"...
Headache: Well, we've had some issues...
Director: O.K., but "pilot A" has been with us for 20 years and hasn't ever had these types of "issues" with anyone else. So, it's like this... When you F with him, then he comes to me and starts F'ing with me. So it's really the same thing as if you were to F with me directly... YOU SHOULD STOP...
End of problem... A=B=C, therefore A=C...

His ability to break things down to its core ingredients was legendary.