To the OP.......Orschlens or MFA will have what you need. Small battery or solar powered charger rated for about 10 miles of fence (that will be hot enough to do what you want). Those will be portable and can be mounted on the side of your coop so it travels with it. The only thing you will need to do at each location is set a ground rod, but you seem able to pound rebar in the ground, so a couple feet of that should do it. You can also attach the ground wire to the metal house and/or hardware cloth that forms the side of the run. Backup.....in addition to, not instead of a ground rod.
Easy way to surround that coop would be to use poly electric rope on screw in or nailed on insulators. Two strands.....one about 4 or 5 inches off the ground.....second about a foot or above that. Low wire prevents them from getting close enough to dig under without getting hit. High wire in case they manage to step over the low wire and try to climb the side of the cage. You can stretch poly rope tight enough to keep it off the ground and prevent it from sagging. Or......if you don't want to attach efence to coop, make your own portable fence using the white step-in insulator posts and set them to stand off from the coop about a foot on fresh mown grass as shown in the photo. Once that coop is surrounded by two low strands of hot efence, , you can set it and forget it and go to bed knowing birds will still be safe in the morning. You were correct about the psychological affect, bit it won't come from being shot at. They don't know what a gun is or what being shot at means. It comes from an encounter with a potent electric fence that jars their teeth out. They understand that.
If you want to hasten their journey to discovery, once the efence is set up, hot and you know it is working, drape a piece of raw chicken skin over it and clip it on with a clothespin. That puts the shock on the end of their nose or tongue. If you have a game camera, set it to record video and point it at the chicken skin. We would all love to see that show!
No amount of shooting or trapping is going to protect you from a horde of unfettered coons, nor is a rooster. When faced with hungry coons on a tear, even the best will die with the rest.