I actually remove my false bottom between uses, and clean it separately. Having failed to do so on one occasion, I have no desire to meet the kind of sour mess I had that next brew day.
I am a bit concerned about it scratching up the sides of the cooler, though, especially since it doesn't quite fit it neatly, instead being wedged in a bit above the floor of the cooler and often with a pronounced tilt. While scratching is unlikely to cause serious issues - it is hot-side equipment, after all - I don't want it to lower it's longevity.