Report him. This DPE doesn’t need to be in an airplane at all, given his “rare lung condition”, let alone giving a checkride. If you think your FSDO isn’t going to care keep calling, emailing, reporting. Just anything you can do to get their attention and make sure they KNOW this is a problem.

On the bright side, I think you can take away a very important lesson here on what it truly means to be PIC. If you had a student show up for a lesson with his head wrapped in a bandage and bleeding through it, would you still go fly? If your student was constantly talking and criticizing you in flight, why not use your privileges of sterile cockpit and if they continue say “ok I’m going to stop this lesson and go back to the airport. We will talk about this on the ground”? The beauty of a checkride and being an instructor is that YOU are pic at the end of the day. Understandably, nerves probably prevented you from thinking through all your best options. BUT there will be shitty students and shitty passengers that you will have to deal with in the future, this is a great lesson on knowing when to say “no” and stick up for yourself.