The Kohler uses a Flushmate pressure assist toilet.
I would buy one first, put it in your unit, and see how it works.
I had one for a while, and sometimes the cartridge would leak.
At night, I would turn off the water supply so it didn't keep me awake.
1 gallon is not very much for bowl wash.
But plug the thing in, and see what it does.
I was kind of embarrassed when someone else used that bathroom.
It never plugged.
That was the good part.
It didn't save any water, it was leaking much of the time.
I used the other bathroom at night, not wanting to wake others up.