You guys are all overthinking this.
The repair is simple: the reasons the dropout is open is that it's relatively thin mild steel.
It's a cheap coasterbrake bike, just use a small sledgehammer and beat the lower edge of the dropout back to the stock opening width. I would also attempt to track down some hardened, 9/16" steel bicycle axle washers with serrated edges to keep the wheel in place against the pull of the engine. The reason your dropout is opening is because you're not using any washers or cheap washers to start with.
I've done this repair a half dozen times. You always note with the owner that the repair you did is only temporary and they should buy a better bike. Also note that you just work hardened the steel you beat back into shape so it might live to see another few months of use But I would get a new frame or better used frame before it opens again.