If you haven’t already, get a referral to a pulmonologist. They are trained specifically in the area of lung diseases and can give you far better guidance than can a GP on dealing with COPD. My wife was diagnosed with pulmonary fibrosis over 12 years ago. This is a progressive and ultimately fatal lung disease without a lung transplant. She was accepted into the transplant program at the U of AZ med school and in anticipation of her transplant entered into a pulmonary rehab program together with a lot of advanced COPD patients. Most of the rehab involved exercise. About half the patients entered into the exercise program enthusiastically; the other half did not. I could tell the difference just by looking. I think a responsible pulmonologist would recommend exercise “as tolerated” for you. Worth a look anyway.
BTW: my wife is now 8 years post-transplant and, aside from some fairly harsh anti-rejection meds she’s taking, is doing fine.