I hope your Dad continues to do well. I don't know about shortness of breath after surgery bur my mom only has about 20 % of her lung function left from COPD. My mom had the cancer but did not have surgery because her lungs were too bad. She had chemo and radiation. Is your Dad under the care of a lung Dr? If not I would strongly suggest it. My mama's oxygen levels were fine without it (machine) and I ask her lung Dr if her stats were OK could she do without it some and he said no, that it would put too much strain on her heart. Your Dad's problem may be temporary, of course I don't know but it may be worth a consultation with a lung specialist. Good luck,
Pat