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.

When you get a perfect squirrel carcass, you put it in your side pouch so it's on you all the time. With the rabbit, you attach it to your horse. After you get both, head to the post office, then take the rabbit off your horse and carry it over to the postmaster and it will allow you to send it. You must have both before you can send it, and has absolutely nothing to do with story progression.