Thoracic outlet syndrome (TOS) is not related to neck origins or instability. Read the section on thoracic outlet syndrome on this website to understand the pathology.
Instability of C1-2 can cause base of the skull pain and headaches but not tinnitis or “tingling lips and tongue”. Instability of C1-2 can cause dizziness.
You cannot determine hypermobility or instability without motion X-rays or a CT scan with the head placed in different positions.
First- work with a therapist who is expert in TOS. You can try brachial plexus injections if this is ineffective. Finally, if nothing else works and symptoms are unacceptable, a surgery to release the thoracic outlet can be effective. I do not do this surgery.