Replace PSP 1004 CMOS battery

I am not sure if this is the right site for a such question, but I found no better option.

My PSP CMOS battery is oxidized. It surely is dead because the console won't keep the time and date. This is not a huge problem for me, the real problem is that I suspect the acid to be the reason of the analog pad working defective (opened the pad and everything seems fine, all the contacts were cleaned with alcohol, plus I tried a second pad that works perfectly on an other PSP and it behaves the same exact way on mine; the battery is located right next to the pad pins, but on the opposite side of the panel).
I'm posting here because trying to remove the oxidized battery all its base is going away with it.
The console is 12 years old but it still works perfectly, except for the pad and the CMOS battery of course, and I would like to avoid ruining it.
So the question is: Is the CMOS battery in the first PSP soldered to the motherboard? Or is it stuck because of the acid and I just have to try harder? What can I do in case it is soldered?

Thank you very much :)

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