I'm not saying I think PBrunsel and Hawkeye are the same person, but I think both of them are lying about who Hawkeye is.
He cannot be an adult with a high IQ who works at a public school. I really do find that part hard to believe...
I don't remember PBrunsel ever confirming or denying the high IQ part, though if he did point it out for me if you would. Anyways, other than the high IQ part, I'd believe that he could be an adult and a gym teacher - his intelligence is probably average or slightly above average, though in his arrogance he thinks he's smarter than he is.