I'm so sorry for you and your DD's loss.
I'm not a big fan of guns at all. But on a farm, they come in handy at times like this. When I was a pre-teen I saw my beloved dog ran over cuz there was a strange dog in the field across the road. It was a blind corner and the driver didn't have time to react. So my dad heard me screaming and crying and ran out from the garage to see what had happened. He sent me into the house to get a rifle, it was just a .22 caliber. I hated it, but it was the kind thing to do in that case. I saw the tire go over his head.
The method of breaking an animals neck mentioned above also works with smaller animals. When we had rabbits for meat, my father would use an iron rod to the front of the head. Chickens were butchered with an old stump and an axe to the neck.
If you get larger animals like goats I'd consider having a rifle. We used to have to shoot opossums after the hens, snapping turtles in the pond that kill the ducklings. And there were the occasional stock animals that we had to put down for humane reasons. Ugh.... I don't like guns. I really don't like recommending them for valid uses. But as you've witnessed, it's much better to quickly end a life that is suffering and going to end anyway.