Right now, butchering and slaughtering animals provides units of meat based upon a simple threshold value of muscle before you can get a single unit of meat.
With the addition of several animals, like rabbits, chickens, and others that are relatively small, many of these creatures are unfortunately not even large enough to produce a single unit of meat. Likewise, meat is only available in an "integer form", how muscular an animal is only matters on larger animals like cows, since animals like goats have to cross an entire threshold value to get from 9 to 10 meat on slaughter.
In order to fairly easily simulate a "fraction" of a meat, why not make a percentage chance for meat to be produced? If something has 90% of the amount of muscle to produce a unit of meat, give it a 90% chance of producing a meat item, and if something has enough meat to produce 9.20 units of meat, make it give 9 units of meat with a 20% chance of an extra 1 meat.
This way, you can't get a full unit of meat from a rabbit, but butchering 20 should at least give you one or two.
Similar methods could be used for the other animal components, so that you'd eventually have a pile of some other things like rabbit intestines.
It would still be reasonable, of course, to have a threshold value before there are any materials that can be gathered, however, such as preventing dwarves from killing a fly and having a million-to-one chance of getting a prepared fly brains barrel.