If you decide to trap the dog, a live-trap would be better in case you catch something else. You can just open the door and let it out, no harm done. If you catch a cat, your own dog, or a friendly neighbor's dog in a leg-hold, you will have (in a lot of cases permanently) injured an animal that you didn't want to. I would either let DH just shoot it, or get comfortable with the gun yourself. I've never had to shoot a dog, but I wouldn't have a problem doing so if my child's safety were at risk. If you're not comfortable with shooting at all, call the sheriff when the dog is on your property, show pictures of it attacking your birds, and by all means tell them about the one that growled at you on your own property.