Ah, a fellow sexy sleeper.
I don't know if the mouthguard will help you with mouth breathing. I suspect not. I tried one of these chinstraps for a while due to snoring (allergy meds and less alcohol did help, however) but didn't find that it helped. But it may help with your issue.
https://www.amazon.com/Anti-Snoring-Chin-Strap-Effective/dp/B07984PQS1
I tried a bunch that kept falling apart, but have recently had luck with the Quiet Panda chin strap.
If you look around Amazon there seem to be chinstraps that look identical in structure but have different branding. Those are cheaper; I'd probably get one of those.
It's really sturdy and won't fall apart. The only problem is that the blue ink dye will rub off into the FFM plastic, turning it kind of grayish-blue, but it doesn't seem to affect skin.
Some type of heavy foam to cushion the strap? At least on the back of the head. Maybe there is some type of heavy shoe inserts that is foamy or of a gel type that would work and can be trimmed. Not sure how to keep it in place to align up with the strap. Might also find some lighter material that can cushion the nose area. Something spongy like maybe made out of the material they make those makeup sponges out of for foundation for women. It is similar to the spongy material they sell as padding for the nose rest on a pair of eyeglasses.
I'm really good at modifying things and designing solutions to things I just wish I were closer to you and I would help you in person. I'm in the San Diego area.
My other suggestion is to entirely change the way you are treating apnea and switch up the mask situation. usually you require a visit to your sleep apnea doctor and they will have a representative that can offer you various options and often when I go in they have the latest inventions in sleep apnea masks to show me.
I don't know if this would be an option due to anatomy and severity of sleep apnea, but I have changed my mask multiple times before I was able to truly get the treatment that I need. I use a mask that has headgear and sits snugly and gently under the nose, but I am thinking of switching to one that is an upgrade of the same mask that also goes over the mouth. Before I do this, I am probably going to try some headgear that someone shared a link to recently that is from Amazon that helps keep the jaw closed, because that is one of my biggest issues. I always put a small piece of clear medical tape over my lips vertically, about three eighths of an inch wide piece of tape from top to bottom lip to just keep my lips together but it doesn't necessarily hold the jaw shut. For me it did greatly reduce the mouth-breathing, though. I don't know the name of the mask right now it's the Dream something or other. I only use the nasal pillow that rests beneath your nose right now, there are no parts that actually go into the nose. It's done okay for me, but my sleep apnea might not be quiet as bad as yours as I have no weight issues to complicate things. The rep from the company showed me an example of the one that goes over the mouth also, and it really is comfortable and not at all obnoxious to wear. It attaches to a separate strap that has magnets that hold it to the mouth part of the mask.
I suggest finding a type of mak that doesn't have to seal over your entire face and get something that just goes around the nose underneath and trying a good jaw strap that will help you. The one I saw advertised and mentioned here, I believe it was under $20. If I can find the link I will add it below, let me take a look...
Here .. https://www.amazon.com/Anti-Snoring-Chin-Strap-Effective/dp/B07984PQS1