I know tons and tons of tosers over 15 years and never heard of a common symptom of nose bleeds. however I do have family that have nose bleed roblems and sinus pain pressure.

I wouold see the doctor and perhaps have a sleep test done and also a cat scan of the sinuses and see if there is a low deep infection.
I had one in the left lower sinus for years and finally a sulfa med kicked it, no more symptoms. My son has nose bleeds, he has very close to the surface blood vessels. You may also. He needed a few caterized...he does have various health issues with seizures, fainting and heart rate. Chest pains, the angina type and he has no TOS, but mitral valve issues. EKG shows problems among other testing.

So I would rule out a secondary health issue going on.
THe docs are still trying to figure out his issues, but one thought I asked the doctor about was dysautonomia. Read this site and see if any sounds like symptoms you may have. It is autoimmune issue.

TOS is bad enough with out the lot that goes with it...myofacial, fybromyalgia, RSD, and many, many have auto immune issues. It is worth reading,
Just as each person gets TOS in a differeing manner or differeing symptoms, this could be something TOSers may want to consider reading about.
I am just starting as an explanation of his symptoms so I am anxious to hear what a neuro doc says....
good luck
di