Lots of good ideas so far. I had another thought, some of the toilets have an open area below the valve that catch debris.
IMO this type of toilet is gross and a bad design. I don't know the exact one, but I know it is a Thetford and is the least expensive one they sell and usually a hand flush. I have seen them go for about $100.
Once we figured out the design of this toilet, two friends of mine brought home brand new RV's and changed out the toilet the first day at home.
If you open the toilet valve and look in there with a flashlight you should see a straight pipe to the holding tank. If not and you see an open area, that may be your problem because there is no way to clean this catch area.
Just a thought.....