Hawkeye is badass, but the problem is that it's nothing but badass. You have to remember that you can't account for whether your son will grow up to be a badass. He might be very timid and conventional. Even if you give him a nice fallback name he's still stick with Hawkeye preceding it on rolls, ¡ob applications, drivers licenses. It's fun to play with but I cannot give it my stamp of Actual approval for a child.
As a middle name he would go by it would be slightly better. I noticed your combo Hawkeye Robert, which is very good and would work well reversed too. The other combo I like the most is Hawkeye Thomas / Thomas Hawkeye.