VTOS (clotting & vein issues) has much better odds for a good recovery.
Barring any accidents or mishaps..post op..
Main factors for a positive VTOS surgical recovery are -
expert surgeon with TOS knowledge. you have that..

If you have poor upper posture, head forward, shoulders hunched or rolled forward? If so working on that aspect may lessen the crowding in the head/chest/neck area..
Some simple posture ideas for at home-


*lay on foam roll or large towels rolled up tightly lengthwise under spine neck to tail bone - small pillow if needed for neck comfort - then just relax with arms at a comfortable position so shoulders can drop with gravity
http://www.oprah.com/own-show/the-ea...e-your-posture

Your doctors should be able to address the blood thinner questions , and how long are they safe.