The crying response to onion vapors has to do with your eyes, not your nose. So mouth breathing won't make a difference.
What Does Work for Onion-Induced Tearing?
Using a sharp knife, to reduce the tearing and bruising caused by a dull one, may help, as will refrigerating the onion ahead of time. Maybe wear swimming goggles? Keeping your eyes closed will do the trick, but you may lose a finger or two!
Wearing contacts rather than glasses while cutting onions can curb the crying, I’ve found. (Those with naturally perfect vision may need to opt for the shop goggles.)