I have never done a cliff rescue involving a K9 and I do not know if our department has ever done it either, so we never have had a need for a muzzle. A ball gag works on our human patients just fine. (thats a joke for those that take everything I say seriously)
The owners said the dog was nice and had never bitten anyone, but like already stated, they can be unpredictable in certain situations, even more so when they are already stressed and scared. You add all the gear hanging from my harness and the noise of it clinking together, and my radio squawking, it could make for a very tense animal. But it was great learning experience, which I can pass on to others as well.