I cannot say exactly what your problem is, but your experience is very similar to one I just had:
One of my three toilets was slow in flushing and sometimes took a couple of flushes in order to completely flush. I checked the water level in the tank, and it was fine. Being that the toilet was over 30 years old, I used the Muriatic acid fix to clear any accumulated lime, even though I did not see evidence of that being the problem. That did not improve the flushing.
Next I bought a toilet auger (aka snake) and ran that through the toilet. I never really encountered anything that felt like an obstruction. Thinking that the stack vent might be the problem, I ran a snake through it. It too did not seem to have any obstruction and the flushing did not improve.
Finally I reluctantly removed the toilet thinking I could clear the line better than with the toilet still attached. Finally I retrieved some hair from the line. It appears that the toilet was going to the tub and the hair was accumulating in the line from the tub and not the toilet. Once I had cleared all the hair I could with the snake, I poured some liquid cleaner that I purchased at Home Depot that was supposed to work on hair and finally flushed it with hot water after several hours. That solved my problem.
My point is that although it is not what I wanted to do, removing the toilet made clearing the line easier and more complete. Perhaps that will solve your problem?