After three power supplies and multiple experiments. . .the Automatic Shower Cleaner is fully functional with an 6VDC 800mA power supply. The power supply is actually putting out about 9V with no load, but it now completely spins.
With the 300 mA power supply it would go a quarter or half turn and stop. The 1300 mA power supply might have worked, except it was actually putting out 5V DC instead of 6.
I've had a couple questions on why rechargeables won't work - typically rechargeables don't deliver as much voltage as a normal battery. Most alkaline batteries put out 1.5 volts, rechargeables are typically closer to 1.2 volts. If one were adventurous, you could modify the case so that it could hold 5 batteries and use rechargeables instead and see if that works.
I know you were all holding your breath, and now you can continue breathing. :)
