How do you mail perfect carcasses for side-quests?

Sounds like you are doing the Wildlife Art Exhibition mission.

If I had to guess, you probably only have one of the carcasses (your post makes it sound like you just have the rabbit). When you have both in your inventory, the option to mail them should become available at the post office (the Wildlife Art Exhibition mission will notify you as well). I also recall skinning the rabbit and being able to send it no problem (I don't think squirrels are skinned).

Some more info here so you don't have to rely on just my anecdotal advice.