Easiest answer, and the way my cousin and I did it, was to learn to shoot with the other hand. I'm right handed/left eye dominant, my cousin was opposite. We learned to shoot both our bows and rifles w/ the non-dominant hand, and have each killed more than our share of animals this way. He has since dislocated his right-shoulder a couple times and can no longer safely draw his bow w/ it when set over 60 lbs, so he now shoots left handed... on all but running shots he does alright, but for running shots you need both eyes open to track correctly and he can't do it well (dominant eye takes over too much).
Oh, we also don't shoot sites on our bows (who needs that crap?) and learned rifles strictly w/ open sites... I didn't shoot through a scope till I was 13.
Many will say try to change his eyes, but why fight it? I feel retarded now when trying to handle a bow/rifle/pistol right handed because I've done it so long w/ my left, even though my right hand is dominant. It also helped me in sports (baseball primarily) by being right handed/left eye dominant... better view of the ball approaching me w/ my left eye... same for my cousin being opposite... we both batted over . 450 throughout our high-school careers.
JMHO,
Josh
ON EDIT: I'd bet good money that his accuracy problem left-handed is because, wait for it, HE'S NEVER SHOT LEFT HANDED BEFORE!!! You can't tell me you were an awesome shot the first time you put a gun to your shoulder and pulled the trigger, why should he be any different? I can't shoot 1" groups w/ my rifles right-handed, but switch it over to my left and they'll be under 1" at 100 yds... weird, huh?