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Shooting long guns dominant eye, especially if taught from an early start position, is the best method. There are a lot of top tier shooters who are cross dominant, and do so. Rob Leatham is just one of them, and he is okay.
Left handed and right eye dominant, I shoot all of my guns right handed. I've done pretty good in multiple shooting endeavors. If he made SWAT, he was doing pretty good the way you taught him. An AR can be run equally well from either side. Shooting both eyes open is preferable for all platforms, so going to the offside eye will diminish that ability. Vision is probably the most important overall aspect of us as a shooter provided we can hold the gun in a proper stance.
My advice would be to try it and at least see how it feels.
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