Please, can someone can help us solve our toilet mystery?

Our award winning Kohler toilet that we installed in our downstairs
bathroom has been acting up. We purchased (and installed ourselves)
this model a few years ago for our new bathroom because we have the
same one in the upstairs bathroom, and it always operated just
swimmingly. The new one downstairs operated just fine, too, up until a
few months ago, when it started backing up repeatedly.

The man from Roto-Rooter can't help us (although this did not stop
him from taking a couple of hundred dollars of our money). He came out
and ran a snake through the toilet twice, and then replaced the tank
valve with a valve that is supposed to allow more water to fill up in
the tank. Neither of these fixes worked for more than a week. He
suggested that for another couple hundred dollars, we should have him
come out and remove the toilet to look for possible obstructions, such
as a bottle cap or something similar that might turn about, thus
causing a blockage sometimes, but not at other times. We decided to
save our money, and removed the toilet ourselves.

However, when we did this we found no objects inside of the toilet bowl
outlet, and nothing obstructing the flow of water thru the toilet bowl.
We took the toilet outside and ran a hose through it and thru the
little holes around the rim, and the water flowed just fine. So,
confused, we put the toilet back, hoping that it would function
properly in the future. But no, it blocked up again, seriously badly,
within days.

I can't believe that there is a blockage in the sewer, as both our
upstairs toilet, which has always worked just fine, and the downstairs
messed up toilet, are connected to the same stack and go into the same
sewer. Also, the Roto-Rooter man said that if there were a blockage in
the sewer, we'd have sewage backing up into the lowest point in our
downstairs bathroom (that being the bathtub), which we don't have at
all. It's only the toilet that's causing trouble.

I'm sure that no one in my household is dropping things into my
toilet that don't belong there. Is my toilet haunted, or what?