To be fair to the stress system, the atom smasher is at the bottom of a pit, directly adjacent the butcher's shop which itself is walled in entirely. So they are only exposed to the guts for 5 tiles, and handle them for 3. Once they toss it into the pit, it's out of sight. Most skins get dumped before they turn, so as to avoid scaring dwarves away from hauling all the meat to the stockpiles. But in the case of elephant skins, it stops the whole operation if they turn before the bridge can come down on it. It would be interesting to let the pit fill for some time, then expose a dwarf to the mess and see how big of an impact a whole lot of reanimated limbs and guts has on their stress levels all at once.
I have magma pumped up nearby, but with the caverns where they are, it will take some work to get it to a good central waste disposal location. Ideally the elephants wouldn't have to be butchered and I wouldn't have this problem. They're being bred into war elephants to be turned into zombies only when they're big and strong, then stored in cages for future use. But from time to time one of the elephant calves is killed by the others.
A more robust trash disposal system would probably involve something like a magma pit that periodically empties, then washes all the magma safe bits to another chamber for smelting or atom smashing, then filling again with magma. I've never tried anything like that though.