It clearly was flooded but this happened before anything was done with the battery. In fact when the battery was being replaced is when the problem was identified as being flooded. Pure and simple this product flooded without anyone ever putting a screw driver on it or anything else
I'm having a hard time reconciling the two statements "The authorized dealer is the one who handled the whole situation" and "I ordered a new battery and when I opened up the case I found out that it was flooded".
The problem is that there is no way for anyone but you to know that the flood occurred before the battery replacement. And floods caused by improper battery installation are not covered. Sounds to me like they believe you tried to replace the battery, screwed it up and flooded the computer, and now want them to replace your computer.
I can see why they might believe that. You said nothing about the computer not working at all, just not syncing up half the time. I have a hard time seeing the computer working at all if it was flooded, even if the battery compartment is sealed off from the electronics. Have you tried drying it out and replacing the battery and seeing if it still works?
If you had truly let the shop handle the whole thing, and they were the ones who discovered the flood upon opening the battery hatch, their word may have carried more weight with Aqualung.