Ok, this idea will make no sense to those who have not gone RV camping. RVs have 3 water tanks — one for fresh water, one for the toilet sewage (known as “black water”) and one for the other drains (shower, sinks) known as “grey water.” When you camp in unserviced campsites for a while you become very aware of the capacities of your tanks.
However, the RV uses the fresh water tank to “flush” the toilet. It seems to me that with a small extra water pump, one could use the grey water, or a mixture — grey with a final spurt of fresh to rinse the bowl.
RVs don’t really flush the toilet, that would use way too much water. You rinse the bowl after #1 and you pre-fill the bowl before #2 and rinse later.
Yes this requires an extra pump, and since you can’t draw from the very bottom of the grey tank, which has some sludge, you even need access to both pumps which perhaps makes this too complex, and thus explains why it isn’t done. And the surplus of “free” water might make you overfill the black tank, which is sometimes your limiting tank. And perhaps if you are going this far you should make the jump to a propane powered composting toilet.