Unfortunately it goes a bit deeper than should I or shouldn't I? You've expressed issues regarding your job in the past, and in some regards there may be kiss up employee's who seem to garner the favor of the layers of bosses.
While the bottom line might be valid and last minute is either a test or real regarding loyalty, to say yes might insure some level of security OR suggest you'll do anything to keep your job. It sounds like the motive of the company has little if anything to do with the employee, only getting a job done, and if your neck is lined up for the chopping block, you can believe so are the necks of others at higher stations than yours.
The fact that there are those around you who don't like you, doesn't distract from your ability, but if bosses listened to every bit of water cooler gossip being spewed, then perhaps they should work elsewhere. While the disharmony may seem superficial, your concern should be in keeping your job, or believing that the next job you apply for will find you have the qualifications and a fair recommendation. "Doesn't play well with others." is hardly mature.