M.R.
What else is going on? I have had kids with fine motor issues, they were always refered to an Occupational therapist. There must be more to the story if you were refered to a Neurologist, especially if they are drawing blood.
If this doctore treats both children and adults, I would be OK with that, but my preference is usually to take my kids to professionals associated with children's hopitals, they just seem a little more geared to children.
M.