People feel strongly about this issue.
For safety reasons, yes, you should buy him a seat. It's unequivocally safer. And probably easier for you (though if you buy him a seat, he has to be in it on takeoff and landing, whether he wants to be or not.)
That said, if money doesn't grow on trees in your household, the odds are infinitesimally slim that there would be any sort of problem that being in a seat would help with. I personally felt like I take risks in my life, and this was one I was comfortable taking.