What a nightmare. I suppose that firstly, your dogs did nothing wrong. It's unfortunate but it's instinct. If a cat like the one you describe came into my yard I would be hard pressed to keep it un-bothered and probably un-maimed with my two JRTs. I'd be willing to bet the shih-tzu would help too. I could call them off and they're quite well trained, sure but they're in a securely fenced area and my house doesn't clean itself.
Secondly, the neighbour is an idiot. You told her you had stock dogs and you put up a fence to prevent issues. Short of running through the neighbourhood every day shrieking at people to keep their pets out of your yard, there's not much more you could have done. That she ignored it and let the cats out is her fault. The yard is securely fenced (I presume) and of a reasonable height. It's her responsibility to keep her cats out of it. As outdoor cats there's an assumed risk. The cat could have been snatched by a predatory bird, hit by a car, killed by a raccoon, eaten by a coyote, killed by a dog wandering into her property etc. She was reckless and unfortunately her cat paid for it. Not to be cruel, but if she cared that much she should have taken your warnings seriously.
If I were to suggest anything I might say you consider adding something to the top of the fence to keep cats off but the only thing I can think up is electrical fencing which I think may be overkill.
I really hope for your sake that after she's had a cooling off period she'll be more reasonable beyond "Evil vicious mean dogs". If she doesn't then the law is still on your side. You're nice to offer to pay for half but I'm not sure that will help your relationship with her in the long run. Personally I wouldn't pay for half the bill if she kept pressing euthanasia. She'd obviously be beyond distressed and well into unreasonable territory.
The whole situation is a nightmare really. I'm compelled to go check my dogs fencing in the morning. The last thing I need is that nasty little dog who runs through the neighbourhood to turn up dead in my yard.