I was diagnosed with Raynauds 30 yrs ago as a teenager! I've been very aware of this fact since I was 6 ,that my hands and feet were always cold. I used to bet my friends I could stand on the hot asphalt longer and win EVERY TIME! I actually have primary Raynauds but secondary Raynauds is due to scleroderma or other connective tissue diseases. I was partially tested for connective tissue diseases but not a complete full workup as my SED RATE was normal and the basic testing was as well.
So yes it does complicate having TOS but it's not the cause. It may put you at a higher risk for vascular symptoms, diagnosis of Vasc TOS or complications.
I would certainly ask the dr to run some labs. It would answer alot of questions and rule out any other possiblities.