If I had shut him down with, "look it up, Oh, you screwed up, sucks to be you, even worse for the girl," then where is the parenting?
That's exactly the parenting lesson he needs. Sometimes you screw up, as he did in not properly reading the syllabus, and you just have to take your lumps. If it were my kids that's exactly what I would have said.
Personally, I hope the girl really did plagiarize your son's paper and the teacher has evidence of that. Because otherwise it's a terrible thing that she was so much more harshly punished than your boy was. I'd consider that to be something of a scandal if it were me. It makes it look like some major favoritism is taking place. I was the farthest thing from a helicopter parent, but if I were that girl's mother (and she really did no more than print your son's paper), I'd be raising a terrible fuss right now. In the absence of any plagiarism, they should have received the exact same punishment.