I advise trying to get the six foot auger.
Looks like you might be in Australia.
If that longer Ridgid toilet auger is not available, or one like it, there is a way to get a regular drain auger safely into a toilet.
You can buy a 3/8 inch diameter snake, the type with a zig-zag type of hollow pipe twisting handle, designed to be spun by hand.
I would get a 25 to 30 foot long one.
Don’t try anything skinnier, it will not work, it will kink up in that big drain pipe.
Then buy a two foot long piece of pvc or similar hard plastic plumbing pipe.
1.5 inch or 2 inch diameter.
Wrap one end with a big blob of duct tape, so it fits fairly snug into the base of the toilet bowl.
Have a helper hold and stabilize the upper end, and rest it on the rim of the bowl.
Or tape the upper end very securely to the rim.
Then run the snake through the pvc pipe, and into the toilet trap and down through the drain pipe.
This lets you get a long snake in there without scratching up the porcelain of the bowl, and without having to pull off the toilet.
You can also buy a 45 degree elbow and push that tightly on the end of the pipe, sometimes this helps guide the snake into the drain hole better than a straight pipe.