Overcoming cross-eye dominance for some is easy, but others struggle without consistent practice. Plus, once your son gets older and possibly evolves into pistol shooting, he will begin to run into even more issues. Try taping over the shooting glasses on his dominant eye with a semi-translucent tape. It will get him used to keeping both eyes open while shooting and still being able to see the sights effectively. Some people quickly evolve with that tactic, but some still struggle, like me. I’m cross eye-dominant and found this to be a huge help, but without consistent practice, the reaction to squint or close the eye comes back.
Being he is only 4, I would possibly have him try shooting left handed. It will be awkward for the first few times, but he may get the hang of it in a hurry, which will eliminate the problem for the future. He’s got 70+ years of hunting ahead of him, might be worth adapting now before his motor skills are carved in stone. I’m a firearm safety instructor and see a handful of kids in each class that struggle with finding the sights until we try off-hand shooting. They will surely say it’s awkward to hold the first few times, but the comfort their eyes get is worth it in the long run as they adapt.
I should also note that I have never owned a left handed gun and I have never found it as a disadvantage in competition, hunting or target shooting. Auto-loading shotguns sometimes have a powder blast, but I always wear safety or sunglasses when waterfowl hunting (like we all should, but many don’t).