TA-088v3 PSPs were the first models that shipped with batteries which had read-only EEPROM access (making them not Pandora-convertable); all newer PSP models have batteries with read-only EEPROM access too.
Note that it's not about just motherboards, you need both:
1) motherboard that can write to battery's EEPROM (TA-085v1 and earlier) and
2) battery that has writable EEPROM.
So you must search for the official battery made in the first half of 2008 or older (because "8C" date code of early 88v3's means "3rd quarter of 2008"). Needless to say, any such battery will be as good as dead by now, so the only realistic option is getting a battery manufactured by a third party.
Unofficial batteries generally don't support writing to EEPROM either, but unlike official ones which have read-only EEPROM to prevent downgrading, they are incapable to write to EEPROM because this functionality requires more expensive EEPROM controller chip and third party manufacturers use cheaper ones to cut the manufacturing costs.
This results in batteries running on nuclear fusion according to their internal thermal sensors, batteries which take hours to drop from full charge to 70% and then immediately die and so on.
My advice is to forget about softmodding the official battery and buying a pre-Pandored one or, better yet, buying a battery with "normal-Pandora" hardware switch.
It's also possible to desolder the battery controller from the dead official battery which can write to EEPROM and replace the controller on a third-party battery with it, but I'd advise against it for safety reasons even if you're good with soldering iron.
TL;DR: forget about it and get any premade "PSP unbricker" battery instead.
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