I thought that grabbing things is what dogs are good at! Can they not make use of "hand-holds" that are made comfortable for teeth to grip with, e.g., a leather cover per steering wheels? (I have little experience of dogs and not at all on board.)
Well, dogs intrinsically have better balance - they're not trying to balance on their hind limbs, and have 3 or 4 point contact with the deck. But they don't instinctively use their jaws to supplement their balance; their response to an unstable situation is to hunker down and lie flat.