Faery's Tale is another kid-friendly RPG that would let you run small animals as PCs. However, like Bunnies and Burrows it takes a more naturalistic approach to its critters (well, as naturalistic as possible in a wood inhabited by fairies, goblins, and trolls).
I guess there's a continuum in these animal fantasies:
Watership Down treats its rabbits in a realistic manner; although they have a language and a rich culture, they are animals. They don't have technology, they can't talk to non-rabbits, they have limited reasoning ability.
Mouse Guard presents intelligent rodents who have developed a civilization and medieval-level technology. They can cooperate with other species, such as rabbits. But they're still tiny mice, vulnerable to being eaten by all the usual predators (hence the need for a Mouse Guard to protect travelers and couriers).
Redwall and Wind in the Willows feature more anthropomorphic critters (throw Secret of NIMH in here, too). They're still normal-sized animals who live apart from and fear Man. But they live pretty much like and think pretty much like humans, use paws freely as hands, make and use tools, etc. Different species interact with and form a society together, even ones that in a Watership Down universe would eat each other. Predator species tend to be the bad guys, which herbivore and omnivore animals tend to be the good guys. It could almost be standard fantasy, except everyone has fur and is smaller than human size.
Finally, you've got Duck Tales or all those Scrooge McDuck comics. Everyone is some sort of animal, but it doesn't seem to affect their behavior, abilities, attitudes, or relationships with one another at all. Species can be a shorthand for personality (fierce, predatory animals are either villains or stern military types). Characters tend to form romantic attachments only with those of the same species, but other than that their "animalness" doesn't make much difference.
Is it just me, or does Martin the Warrior's nose look awfully big in that dramatic painting? (Not that you'd say so to his face!)