That is part of what is probably going on. However merely changing the potential once wouldn't accomplish much as even just a few positive copper ions would have enough charge to negate that -0.44v. I suppose if you had a large enough chunk of iron then you might have a large enough well of electrons in the conduction band. However my understanding is that this would be an outrageously large amount of material. How big exactly I don't know. Perhaps as large as a mountain? It would be interesting to quantify.
If you think about it, you can change the potential of even a great thick chunk of metal in the blink of an eye just by briefly contacting it with one terminal a little battery. Only the tiniest amount of material in the battery has changed states in order to do this. So it doesn't take much to change the potential of even a large chunk of metal. So we see that the inherent capacitance of the conducting sea of electrons in the bulk of a metal material is quite small, thus for a practical set up we would need a continuous loop of current in order to plate a visible amount of material.
So, how can the iron object replenish the electrons it donates?