So this is a topic I have a lot of experience with...currently I'm still teaching Abi not to eat our 3 cats. LOL
Muzzles are a mixed bag for me. I think they can be a very good safeguard, and I have used them with particularly predatory foster dogs. It's just a bit of insurance that, should things get out of control, they can't injure the cat.
However, I often see them used as a band-aid. What I mean is, people think that because the dog can't bite the cat, things are OK. But the dog is still growling at the cat, chasing it, agitated by it...if it physically could, it would still bite the cat. That to me is playing with dynamite. It's only a matter of time before the cat and dog get out together, and seriously it takes 2 seconds and the cat is dead when you're talking about a dog the size of the average GSD.
In fact, creating a physical barrier like a muzzle or a see-through gate (like a baby gate) but not training the dog to ignore the cat can actually make the problem worse. Frustrating a dog tends to increase the dog's drive. Seriously, this is an accepted training method--ask any working/sport trainer who to make a dog toy-crazy or increase their prey drive, and I guarantee that frustration will play a role in it. Frustration makes the vast majority of dogs want to go after it more, whether "it" is a toy, a cat or a person.
So yeah...big mixed opinions here.
You sound really committed to Beth, and as long as you train her as you would without one, then I don't think muzzling as a safeguard is a bad thing, if she accepts the muzzle. But I also don't think it's going to fix anything and training is the key component there.