The pokedex specifically mentions it can't breathe unless it's underwater. Maybe really powerful lungs would explain why it doesn't drop dead 5 minutes after being sent into battle, but unless the lungs of the draco part of the fossil exists in its butt (lol) where would the lungs and windpipe be? Is it even alive at all?? Is it just a zombie??? Help???
Pokemon doesn't always make sense. Dracovish is a mashup of two different creatures, it should never have been alive in the first place. How can Magikarp, Feebas, or really any fish Pokemon survive out of water? How can Dodrio learn Fly? Game Freak doesn't always make sense, which is the case with Dracovish's breathing methods.
Hope I helped!