>Head gear is needed.
Unless you exhaust your breath outside the bag. For instance, a swim snorkel in your mouth - which would take a little training to use, but people with sleep apnea learn to use a CPAP machine all through the night. Ideally, there'd be two check valves so you breathed out one tube and in through another.
If you take that step, you've got the possibility of saving even more heat. Because typically we breath in ambient, cold, dry air and exhale warm, moist air. The breathing passages in your nose help to retain heat and moisture but an external heat exchanger could do even more.
Aside: absolutely try to train yourself to nose breath instead of mouth breath - you'll retain more heat and moisture (and moisture in air IS heat).
There are heat exchangers and difusers sold for skiiers and mountaineers. They each want them for the larger surface area in the event of being buried in an avalanche. Additional, the HX function retains more heat and moisture - a big issue at 25,000 feet where you're always dehaydrated and liquid water is hard to carry or melt.
For BPing, the added surface area doesn't help, but the HX and moisture retention would. Simplest would be exactly a swimmers snorkel with some sorbant packed in it to capture heat and moisture from the exhaled breath and transfer it to the inhaled breath.