I would first make sure that shooting for a 10 year old is fun and not a chore just because a bunch of adults think he needs to switch hands to shoot better.
Eye dominance varies in it's degree and may not be a factor in archery if the shooter is consistent in form.
Eye dominance can change with time especially in cases of those who are nearsighted, wear glasses or contacts or have lasix surgery. Lasix surgery 30 years ago definitely affected my eye dominance.
Some kids are just naturals at shooting sports. I took a friend's son elk hunting one year and beforehand we practiced at a 3D course. The kid (18yrs old) is left handed, shoots a left handed compound and is extremely accurate. I was shooting a right handed longbow. Half way thru the course he asked if he could try my longbow. On a standing deer target at 23 yards he put three arrows in the vitals like he had been shooting right handed his entire life.
The other option is to buy your son some golf clubs. Tiger Woods and Jack Nicklaus are both right handed golfers and left eye dominant.