Far too much money being spent on plumbers.
Just pick up a Toto Drake and send the old one on it's way.
There is something in the trapway that is hiding, and if the plumber can't find it, it's time to give up on it.
I had an American Standard Cadet 1.6 gallon like that.
Once it started having problems, it never stopped.
I too, took it outside, and augured and augured.
I never did find the problem.
We use a closet auger with the large 1-3/8" end on it.
Smaller cables can go right by the obstruction and not move it.