I had a vest for chest infections for many years but never used then daily. I also had one of the early inversion tables or hang ups. The ability to return to vertical quickly is quite simple and as fast as you need. I never thought to combine the two and there is enough good reasons to ask a respiratory therapist of doctor.
We aren't exactly straight plumbing when you invert the body. Stuff draining from the lungs would fill the nasal structures in the head, bypassing the mouth. Even a normal productive cough shoots lumps of mucus up above the roof of the mouth in the back of the throat. A totally full nasal cavity sounds like a problem that you may not want.
What strikes me as the ideal method is a compromise of inversion with a vest until it's time to cough and swing back vertical to cough. A concern I have about inversion is the increase in blood pressure in the head and the weight of the heart and organs above it weighing on it. Having said that, inversion did little for my back but feeling better from having my organs and body fluids like the lymphs and blood a chance to get repositioned and stimulated was profound. As for reflux, full inversion may not work, or be needed. My GI doctor recently remarked that most people have a non functioning upper stomach valve after age 37. If healthy folk have reflux, it's going to be an individual choice depending on how much it interferes.
Try before you buy, everyone knows somebody with an inversion swing. Long before either were invented, I gravitated toward the plow stance by luck. In fact I had a ten pose routine recommended by my brother, who was immersed in yoga when I was 9. It was great, in ten minutes I was limber, my gut got a thorough massage, lungs were hyper inflated, sinuses loosened and muscles were warm.
With your respiratory therapist/doctor's blessings, or cautions, let us know how it goes.
LL