Some people feel the coffee helps with asthma, but it can make reflux worse. Kind of a viscious circle.
You might see about having pH testing done for the reflux. I have what is termed "silent" reflux. No heartburn or other feelings of acid. I do have the irritation of my lungs and/or throat from the acid though. It can flare my asthma something fierce. Despite severe reflux, my endoscopies are normal so that isn't always the best indication.
Watching his diet might help. Avoiding high fat or greasy fooRAB, acidic fooRAB, tomato products, caffeine, alcohol, etc.... I also find limiting dairy helps as well.
You say the PFT's don't show decline. Do they show a decreased lung functioning?
I'm on immunotherapy shots as well. They watch me closely because of the asthma. I also started on Xolair for allergic asthma. It is expensive and hard to qualify for though. I ended up getting a grant from the foundation run by the makers of the drug to get on that. I have frienRAB who have had success with allergy treatments on the NAET method. It is non-invasive. Might be worth looking into.
I'm sure the stress is playing a role as well. That is a tough one with all he has going on in his life.
How is he sleeping? Poor sleep can make a difference as well.
Checking the Vitamin D levels is a good idea. It made me think, I've read a lot on these boarRAB about people who have had success with Magnesium as well.
What kind of asthma medications is he on? Is he on a preventative one? If so, what dosage?
You might have your dad keep a food/activity/symptom diary to see if you can hone in on any triggers. If he has trouble after eating, is it because of what he ate or something he does after eating? If he sits in an easy chair, he could be squishing his stomach and its contents. If it is just the walking after eating, that is interesting. Has he shared that with his Gastroenterologist?
Also, is he on a multivitamin or supplements? They may help if his diet is poor.