I don't have much experience with cribbers, there is only one at the barn that I board at and I don't believe anything is done with her to stop it. From what I've heard, she made herself get nasty neck wounds from wearing a collar and trying to get it off. So I can't really relate in that way.
However, if the BO doesn't want the electric fencing on the gate, there's nothing you can really do about it. We hate electric fencing on nearly all of the gates at my barn and it isn't a problem. I used to feed out there at nights a few times a week and never had any problem getting hay in and out or whatever else without being zapped.
The muzzle does seem to be affecting your horse negatively, though, and I personally wouldn't want my horse to be forced to wear something that is unnatural to him 24 hours a day. I would suggest, like many other people have, using the anti-crib stuff on the gate and see how that works. And if that doesn't work, I would seriously consider moving barns. My boy would never have to live with a muzzle on all the time.