I think a 6V/6A power supply will melt this toy.
Brand new AA alkaline cells have an internal resistance of 0.146 ohms. So 4 in series have 0.584 ohms plus the resistance of the contacts and wiring. 6A through 0.584 ohms equals a voltage drop of 3.5V so you would expect the toy's designer to allow for it, and have the toy operate from about 3V.
With the heater operating at 3V, its resistance will equal the battery's internal resistance. The battery will heat as much as the heater!
The current with a brand new battery is 5.14A and the power in each the heater and the battery is 15.4W.
How long would AA alkaline cells power the thing? Looking at Energizer's curves, only about a half hour. Of course the power would begin dropping immediately and after a few minutes wouldn't be enough to solder anything. Sigh. :lol: