Yes, it has to be closed for the air pressure to be correct for bipap, and of course will cause dry throat and mouth. I know its all so much to assimilate, and sometimes nothing stays constant long enough to get a good handle on it before something else arises, but, if you can work on getting bipap straightened out, it will help Bruce feel better all the way around. You are correct that his headaches may be coming from inadequate air exchange. Most of the time is is a build up of co2 that causes the headaches.
I know tomorrow is a Saturday, but consider putting a call in first thing tomorrow, to the provider that set him up with the bipap equipment, and explain that his mouth does not stay closed, even with the chin strap. Explain he's having headaches, and ask if they can get a full face mask out to try asap. Depending on your circumstances, it might speed things up if you, or someone you can ask, can drive out and pick up the mask. They are bound to have something in stock at their storefront. When I've had problems with my machine, it invariably has fallen on a weekend, or holiday, or, last time (as an extra special bonus for us all) the night after I had surgery, on a weekend, when the RT we deal with was several states away, and the guy covering for him could not be reached. How fun is that! The pharmacy was closed, and our RT actually called and had the pharmacist drive out and meet my son there to give him a new machine for me to use. I'm telling you this to help illustrate that home medical equipment companies are accustomed to working with patients that have needs outside of regular business hours.
Don't let the term "full face" put you off, it just means the mask fits over his nose and mouth, so it doesn't matter whether or not his mouth is open or closed. Many bulbarians need to use this type.
Good luck, stay strong.