Both of my daughters are Left eye dominant. The older one is lefty for writing and other fine motor skills but is right handed for sports. When she started shooting she went straight to left hand without me saying anything. I bought her a LH Savage and she's good to go. The little one on the other hand (pun intended), is right handed but left eye dominant. I still haven't figured out what is best for her. I may put her behind a LH rifle and see what happens. She shoots well right handed but does take too much time looking through scope. Myself, same boat left eye dominant and right handed for the most part. Dad started me right handed and that's my comfort. Stinks though that I need to close left eye when using a shotgun. Sometimes eye fatigue hits during long range sessions due to closing it.Both my sons are left eye, right hand. They are 6 and 9. I am still figuring out this question. I believe I am going to end up with 1 of each, a right and left shooter.
Neither kid can easily control/wink his eyes well. The 9 year old is worse and I have him working on shooting lefty. The 6 year old can control his eyes better and wants to shoot righty.
Have him shoot what feels natural I think.
OP - I would direct your son to left hand.