I have made one before, me, me, me!
You have made an aluminium oxygen battery, the copper plate acts as an uncreative electrode and plays no useful part in the reaction. From experimentation and what I have read, it is possible to extract the aluminium from it's dilute oxide sludge when discharged into the water. I discharged my aluminium battery for a while, measured a voltage drop due to depletion, I the applied a reverse voltage for a few minutes and noted a voltage increase of about 0.10 volts. So, yes you can recharge these, the voltage per cell when working with pure water makes their use in most applications absurd, I obtained 0.25 to 0.5 volts, depending on battery configuration and so internal resistance, electrolyte solution etc. You will need at least 3 volts per cell to recharge, more if possible. I noted voltage increase with a lower voltage recharge however. Try the 12 volt charger!
Remember, when passing an electric current through salt water (an H2O, NaCl solution) you will obtain hydrogen from one terminal and more importantly CHLORINE of of the other, poisonous, not good for humans, stay away. Of course, the risk is minimal, given good ventilation and low concentrations. I would have done more work on aluminium oxide electro chemistry, very ingesting, the ability to create a rechargeable, safe cell, but a lack of equipment prevented me from obtaining any more useful data.
The confusion with cryolite is that it is not strictly necessary for electrolytic extraction of aluminium for it's oxide, it just reduces costs. The aluminium oxide is suspended in water, allowing ionic movement, hence, there is no need for the heated cryolite used on the Bayer process.
I hope this helps,