Alright, so you do have the animal meshes placed into your map? There should be an alternative file for these. Like Animals.fho or something, that is exported.
The easiest way to explain this is;
Make sure that the animal meshes are NOT grouped in with any of the default groups in Map Maker, like the MyRocks, MyTrees, Miscellaneous, etc. They need their own group.
Once you've exported the map, go into your exports folder and select ALL the files that the Map Maker produced, and copy them into the folder you are wanting to zip. That includes all object, sky, weather, map, etc.