We have low water Kohler toilets that can back up easily. And there is a trough and slot in the bottom, which prevents using standard plungers to clear backups.
To prevent a metal snake/auger from scratching the porcelain, I got a length of heavy gauge plastic tubing that was flexible enough to shove well into the toilet's S drain, while the back-end remained about a foot or two above the backup water level. The inside diameter of the tubing was larger than the snake/auger head which was not very large..
Using care, feeding the snake/auger into the tube and further into the toilet drain, the blockage was cleared, (and on more than one occasion). Have a rag or two on hand to wipe off the snake/auger when retrieved.
Blockage was gone, and the porcelain remained scratch free.
Also, the auger used was an inexpensive hand crank model.