I think the blown fuse is for the battery, so its possible a faulty battery has drawn excess current from the charger.
Have you checked the battery electrolyte?
At any rate ask your dad to do a current check for the battery charge, once the charger is replaced or repaired, assuming its faulty.
The blown fuse could be a one off, or there may be a problem.
Rather than use a multimeter which is normally only rated at 10 amps, use a claw ammeter, or better still i have one of these.
http://www.maplin.co.uk/automotive-current-tester-217879
Bailey do not recommend that the charger is permanently powered, although i leave ours on for a two week tour, i never leave it permanently powered at home.
These units run hot, and are housed in a poorly vented enclosure IMO.
We take our caravan in shortly for service, the caravan charger will go on power the day before, that other than say a days charge every month is all it gets through the winter, the battery is soon 8 years old, and still performing.
So i would have a word with your dad about your charging schedule, i don't believe in leaving electrical appliances powered up when not required.
PS, a melted fuse implies a overload current rather than a short circuit.