I think that's a good idea. Just make sure you go to a reputable place (we took both kids to a major university). Then you can take the report back to the school as an independent evaluation, and they have to consider the findings/recommendations.

Personally I think everyone with ADHD probably also has psychomotor OE (at least hyperactive type ADHD). I wouldn't rule out ADHD. The focus issues and impulsivity are major flags.