I agree completely that it can be fixed. But if it's surface mount components you need an expensive soldering station and lots of practice. Not something you can do with a Radio Shack soldering iron and some pipe solder you had lying around..
There was a picture posted somewhere that showed it was not surface mounted components right? Maybe in a link to another thread or somewhere.
It's worth opening the shutter if it's the kind of work you're able to do.
Leaky cap. or device is certainly a possibility and they're often unmarked - which can be a pain.
Often though current drain is via corrosion products, look for green goo/whitish streaks anywhere in the circuit but especially around the battery box.
Without an accurate lab. power supply & a dummy battery you can use a thin metal disc/insulator/metal disc, a wire soldered to each disc, between battery terminal & contact to measure drain with a multimeter on progressively lower milliamp ranges. Checking the drain at all states and speeds of the shutter may help diagnosis.
Good luck with it!