I'm no expert, but that does not stop me, so here are my thoughts.
My son is also right handed but left eye dominant. But in his case, he struggled with this (lefty vs. righty) when he started playing tee ball. Actually, until he was 3 or 4 we thought he'd be left-handed, but slowly, he did more things like drawing right-handed. Voila, he's a righty like his Dad (my father was LH and both my brother are). But in tee ball he struggled to hit RH and LH just seemed a little smoother. A week or two in he stuck with LH and has never looked back. His RH swing is sad to see.
Given your son's age and lack of size, so far, and given he makes good contact as a RH hitter, I'd be concerned about making the switch so late in general. But I'd recommend having him work with a really knowledgeable and experienced hitting instructor and see what he thinks. Good luck.