I'm so sorry!

In my opinion, the gash looks large enough that putting her down is probably the best answer. It is always so hard when our pets get injured or sick and we have to make that difficult decision. I hope your cousin and you can work it out so that it doesn't happen again. Once when we had neighbors, one of their dogs jumped our run fence (roughly 6 ft) and wiped out the entire flock. After that we strung electric wire around and didn't have any more problems. Would a similar setup work for your situation? Or are they free range? Again, I am so sorry for your loss and your injured hens!