Keeping the level only half full is making things harder for you, but can you drive around and make some hard stops to slosh things around without causing stuff to leak? This is should be done after leaving water in the tank for a day or so to soften stuff.
Also, assuming you can turn off the water, open the valve, and see the stuff down your toilet with a flashlight, you might be able to stick a metal rod down there and sort of stir it to break up the stuff. Then drive around to slosh things, and dump again. But you need to keep water in the tank after dumping or the stuff will just harden again. (Don't use a stick or anything else that might break off and cause more of a problem.)
I never had a pile of poo, but I have twice had a pile of toilet paper after grandchildren and other guests stayed with me and used too much of the stuff. It took several days of stirring and sloshing to clear the mess up.