I'm super strong left eye dominant, like your son. I have played sport all my life from basketball to football to golf to baseball all right handed, left eye dominant. I place forks, spoons and knifes in my right hand. I use my right handed mouse on my computer. I use my right hand with the TV clicket, a coffee mug, etc.
My brain KNOWS how to compensate my right handedness with my left eyeness, it does it every waking minute of my life.
So does your son's.
Now, why would you change up his hand/eye/mind coordination with switching him to left handed bows ? Are you going to make him do everything left handed ?
I'm a pretty good shot, better than most to be honest, Howard Hill, Larry Yein .... both cross dominant. I've argued this for ages, and still am in the belief that since I do everything else cross dominant, its MORE natural for me to shoot a right handed bow too.
Just my thoughts on the subject, which differ from most I know
Brad