You take it to the vet. The vet either agrees with you and puts it down humanely or takes it off your hands. You don’t take it to “the gravel pit in a fit of anger and shoot it.” She is a horrible person.
“You take it to the vet.” Why? Shooting a dog in the head is instant death for the dog.
I would take it to the vet because it isn’t legal to fire a gun on my property. But on a ranch?