Hi TaraMa
It is a very interesting question. Sorry to sound negative but I think you need to be realistic about whether physiotherapy can help dyskinesia. As far as I know there is absolutely no evidence for physiotherapy for dyskinesia and given the the pathophysiology of dyskinesia in the PD patient (usually levadopa toxicity) I think it is unlikely that you can do anything for it. At some point I think there was an interest in biofeedback for dyskinesia but was found to be wanting. Sorry - couldn't find any reference. There are things we can help patients with PD with but dyskinesia isn't one.
RE: a mouth guard if the patient is experiencing bruxism due to the dyskinesia then an appropriate mouth guard might protect their teeth. However the complex and large range of motion of the disorder might mean the mouth guard becomes dislodged which could cause the patients problems and there could be a safety issue here. You could certainly trial one but I would keep a close eye on how it goes - discuss it with a dental colleague - preferably one with a knowledge of neurological disorders.