Try the charger again on the 12V battery. If it's running cool, then it's the power supply. Some power supplies don't provide a good, clean source of DC voltage. A "dirty" supply will cause the charger to operate incorrectly, which could easily lead to overheating.

Now, if the charger still gets hot when you use a 12V battery, then we'd agree that the charger has a problem.

Just some things to consider.