I have see the battery leakage on your mobo, it's not bad. Simply clean the corroded area with some isopropyl alcohol, melt and suck the solder (mixed with battery leak) out of the board.
Them, with a scotch brite (or a flier), rub the area (carefully, don't get near traces or chips) until you see the bright copper and pour some solder on it. Clean everything with a soft and dry wipe and that's it.
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The PSU problem: you can always swap a dead Amiga PSU with a AT peecee one. The only problem is the missing of the "tick" signal.
The A3000's PSU pin assign is found on
pinouts from Russia.
The peecee AT PSU pinout assignment is
here.
The only one thing you need to modify on the Amiga itself is move the J300 strap (jumper) to position 2-3 (internal tick).