Had there ever been a study comparing a vegan diet to a paleo or gaps diet?

No good ones: So many other lifestyle variables accompany food choices that to get anything other than garbage, people would have to be randomized for a few decade to different diet and followed prospectively. That's not going to happen. Your best move is to choose your food as you like, avoiding the very sugary, greasy, and salty stuff, and exercise intensively as your body was made to do.