By sustainable I mean its impact on the environment, animals and humans. I don't mean how profitable or easy it is.
Rabbit is one of the most sustainable livestock. It requires less food to produce more rabbits and they don't need much land. A single female produces ~50 kits per season, taking 8-12 weeks to come to market. Ironically, they aren't considered livestock by USDA, so you can skip most of the red tape. As far as feed goes, there are many options that will depend on the farmer; pelleted feed is the best but most costly, so you can mix in either foraging or supplement with various other feeds like different hays, oats, etc. Whether it's economical depends on a number of factors, but there's over 3,000 rabbit farms in USA right now.
Same with goats, very sustainable. In many places like islands, goat is the preferred livestock as it requires less land and feed. And obviously you can forage goats in places most animals won't since they'll eat nearly any plants.
Pigs (and other livestock) historically were raised through the year and only slaughtered in winter. It's the last 100 years that has completely changed how and when people in the West eat meat, people's assumptions about how we must farm, how we must eat, etc. Our diets don't have to remain the way we are. For example, since Chinese people prefer to eat pork, they actually have half the world's pig livestock. We like beef so we have a lot of cows. It could've been reversed if our cultural tastes were different. Similarly, we could just eat less meat, and our tastes would develop towards the whole universe of non-meat foods.