I rather doubt that the bad battery caused the converter to die, though I suppose it's possible. The PD is supposed to be protected against short circuit output conditions and open circuit output and most anything between. Maybe if it was operating at full output for a long time with inadequate cooling the fuse would eventually die of "old age" on its own absent any real fault.
The fuse should read zero ohms (or close to that) whether in situ or not. If it's open, it is blown. From your description it sounds as though it's probably the main AC input fuse, and that would usually imply that something is short circuited in the input side of the converter. Converters these days are basically just largish switching power supplies, so check the usual input side things--the big electrolytic capacitors, the input rectifier, the switching transistor....something may have failed short circuit.
