Unfortunately I'm not a clinician that could speak to ways of addressing this. Some people definitely do seem to have pain associated with stuttering. Though it surprises me (again not as a clinician) that relaxing actually increases your pain, because for most stutterers, reducing muscular tension helps with almost all symptoms of stuttering. I think you're correct that there is little to no research on this specific question (maybe case studies at best). A major challenge in stuttering research is the small subject population we have to work with, let alone subsets of that population who have specific manifestations of it.