Long term raised rabbits are better than wild rabbits for feeding whole prey. The game rabbit does not have near the fat content of a caged rabbit. One the game rabbit must forage, cover ground and when spooked will run like none other. Those three things keep them thin and just about fat-free. The caged rabbit is fed a diet (rabbit pellets) that is readily available, he only travels from one end of the cage to the other and more than likely will never break into a dead run. The lifestyles are different and the caged habitat will lead to a fattier rabbit.
Same with chicken. A chicken house chicken is fattier than a yard chicken who has roamed, scratching and pecking his whole life.
Deer the same. Deer hunting has exploded into a huge sporting venture. At 45 (not 145) I can remember deer absolutely having to have beef fat added to make deer burger because it was such lean cuts. Today thousands upon thousands of pounds of corn re poured out and the deer congregate in that area as the food is plentiful and easily accessed. When we skin deer now there is a fat layer that was not there as a kid (in rural NC, maybe different in other parts).
Granted the average chicken is a better choice over the average rabbit but a tame-raised meat rabbit will not starve a dog. His living in the wild counterpart may, but there is a big difference between the two. EWO