Sounds like you have a duff battery to me.
That would explain the battery not holding a charge.
The battery acts as a voltage regulator too, so if it's knackered, it could be allowing too much voltage through, which may be why your fuse is popping.
The first (cheapest) option though is to check out the wiring diagram, then follow the wires from the fuse to wherever they go. (I forget which the 3 amp fuse is for, indicators maybe?) and hope you find a bare wire or something obvious. In fact, it wont do any harm to search all over the wiring loom for any faults.
You could do with a voltmeter to check the state of the battery. Then with the bike running, check the voltage again. It should be up over 13 v with the engine running, as the alternator charges the battery. If it's not, then your problem is probably in the charging circuit. If it is charging, then your battery looks iffy.
Good luck.
Baghira 660 for the road, CCM 404 for green laning. Singles rock!