I would do the following with battery leakage suspect:
- Remove the module
Remove the LCD assembly (panel, zebra strips)
Clean with 99% alcohol and a fine brush the entire module.*
With a damp cloth (dampened with water) gently clean the LCD panel and zebra strip contact points.
Clean the screws and case.
Reassemble
* I take a natural bristle brush and clip the bristles short so they are about 1cm long to ad stiffness. Normal keep moisture away from the trimming capacitor but since you are using 99% alcohol (local pharmacy) not to worry. The object is to remove all the battery leakage.
I take it there is no blue precipitate, just clear oily liquid.
Cheers, Geoff