Yes, I think he should still be considered left-handed as that's his dominant hand for writing. I think many naturally left-handed people have to adjust to living in a right-handed world, so they've learned to use their right hand to manipulate many everyday objects. Scissors for instance - I'm strongly left-handed, but I use my right-hand for scissors. My mum was exactly the same. Back in the day when she was a child, I highly doubt left-handed scissors even existed so she compensated. When I was a child first learning to use scissors, I no doubt learned from my mum, so I'm right-handed too. Same idea with a computer mouse - most people are right-handed, so I learned to use a computer mouse with my right-hand. In a way, that's actually a good thing because it frees up my left-hand to write with.
I wonder if your son is left-side dominant? There's tests you can do to figure out whether you're left-sided in regards to footedness, eyes, ears. I'm strongly left-side dominant.