There was a boy in our son's kindergarten who transitioned from Montessori to public school and did fine. However, if you're happy with Montessori and have no motivation to change, I wouldn't make the move.
The bigger issue here, though, is what you're doing to address your son's symptoms of ADHD? Has he been evaluated by a doctor? Are you seeing the extreme hyperactivity and limited attention span at home, too? He's at the age now when you need to take this seriously. If you're limited in what he can do, that's a red flag worth paying attention to and it's time to do something.
(ETA: A pediatrician should NOT do the evaluation or be advising on this condition. You need it done by a specialist like a child psychiatrist or neuropsychologist. I'm not sure why the wait until six? AAP has approved assessing for ADHD as young as four. A skilled specialist will give you a definitive yes or no. They're experts in identifying this condition or saying, "No, just neurotypical." Also, if your son does have ADHD, I would lean heavily toward public schools. They're in a great position to help kids with special needs as they're required to by law. In our personal experience, private schools are far less tolerant of conditions like ADHD.)