There are contexts where a neurologist could be relevant (e.g. sudden onset stutter), but for regular stutter, it's likely not going to yield anything of use. Perhaps they wouldn't even know what to look for, since there isn't a solid and substantiated theoretical framework in place that dictates what exactly stutter implies neurologically, if anything (in terms of usefulness). But it's always a good thing to check in with doctors to see if something's going on that could be treated. It's just not a common route for stutterers, since our thing is more peculiar.