I was thinking that you could combine a plunger with flushable cleaning pads/wipes. This way, the bowl gets cleaned as it gets plunged.
I am sorry that I am not a better artist, but the internal and external pads could be connected by a string so that when the cleaning is complete, both can be easily removed and flushed. The extrernal pad is a conical disk shape with a hole in th emiddle for the plunger to go through. The internal pad attaches to the bottom of the handle which would protrude a few inches into to ball of the plunger.