Sorry to hear about your struggles, we have all been there.

From my personal experience I would not delay the surgery if testing is showing you have TOS. Also PT is not recommended for vascular or arterial TOS which I had bilaterally. Mine was congenital so I am not sure if an accident can cause that but for this type of TOS the surgery has very good outcomes if you have a good surgeon. Have they tested the blood flow in your arms? CT or MRA with arms up?