Even if we had a gun, AND could learn to shoot accurately, we'd have to be waiting when the fox shows up. And get a clear shot. It just doesn't seem practical...
We have two dogs, and I'm sure if they saw a predator they would go ballistic noisewise, but again both these killings happened without them alerting us.
You don't have to learn to shoot it accurately. Just poke the barrel between the holes in the live trap and pull the trigger.
What kind of dogs? You could let the fox out and let the dogs get it.