I know someone who had used the sleeve from old oversized cotton t-shirt. Cut it out around armhole of shirt, and it stretches enough to make a sling thingy for the chin. One edge is larger around than the other, (shorter edge to front of face) so it cradles the chin, hence holding the lips together too, Could be dorky looking, but then, who cares ?
I think full face masks do allow for mouth open, no strap needed. I use nasal mask, my mouth stays shut though.