My first thought is that you can't do as much from home as you think, if the problem is at school.
I'd ask the teacher if the school psychologist or other learning specialist can sit in and observe, to see what's happening when he goes off task? Schools provide intervention and assessment services free of charge.
It could be anything from distraction by another kid, anxiety, eyesight or hearing problems, sensory issues with a noisy classroom, and on and on. You need another set of eyes in there. And not yours - you'd be a distraction, an embarrassment (any kid's parent would be), and a non-objective source.
What does it mean to "sit out on lunch"? He's not getting lunch time? Or he can't sit in the cafeteria with the other kids and has to eat at his desk? No lunch is NOT okay. I actually think recess is necessary too. Maybe staying after school to do some work is possible. But also another objective observer might determine that he has issues (processing, boredom, who knows?) with certain types of work, and maybe there are modifications that can help.