If a dog is on your property, mauling your animals, there's no way to really handle it nicely. If it happens a second time, there's no peaceful resolution, the dog needs to be put down, and if you catch it in the act, all you have to do it is shoot it, then take a picture of the body on your property for proof. The animal just has to be on your property, and a threat. Since there's proof of a previous attack, she's 100% justified if the dog steps on her property again, to stop it with any force necessary.While its perfectly within your rights to file a formal complaint since your neighbor has offered to pay for the damage i wouldnt. I would go over to their house and explain to them that while you will take the money and appreciate the gesture that it cant happpen again because it cant replace the time and love you put into your pets and it doesnt really solve the problem. If you can, let them know your recording the convo on your phone or something as documentation that they have been warned incase it happens again because if it does you will file a complaint or if you feel threatened will shoot the dog. But dont record them without permission as that is illegal. If they say no to the recording try having them sign something. If they wont do either i recommend investing in a camera system or game cams around your property so if it does happen again you have proof the dog is running around and that it killed your pets so when you call animal control you have solid proof. Each time the dog runs free whether it comes it kills your pets or not and is on your property and yoy show it to animal control they will get cited amd have to pay a fine and if it still doesnt change after so many times they will remove the dog. But try a peaceful resolution first and let them know you dont want a bad relationship with them but you will do what is necessary to protect both your pets and family.
In Texas, killing livestock is almost worse than killing a person. Fines would be the least of the person's worry. You can get jail time if you knowingly allow a livestock killing dog to be free to kill your neighbors animals.