Hey,
You have to use a 3.6V/3.7V battery, it does not matter about how much mAh there is on the battery, only issue you have is that you can not use anything higher than 3.7V or lower than 3.6V, otherwise you're looking at either melting/overheating/possibly burning your PSP.
I've done this mod using a simple 3.7V 3300 mAh battery (Lithium Polymer), I used the battery surge board from the original PSP battery then attached it to the new 3300 mAh battery I bought, it work's as it should no overheating etc, only downside is that you will get a inaccurate reading of how much battery life there actually is on the new battery (again a PSP hardware problem, no sensor).
Overall try to buy a lithium polymer battery which is 3.7V (they all sell this online or at you're local hardware/electronic depot) and think about how much mAh you actually need, the higher it is the fatter/longer/larger you're battery will be, the more complex it will be to attach into the PSP.
My battery size was 7cm by 7cm by 0.4mm (WxHxD(thickness)), I used the UMD compartment and cut a bit of plastic from the interior and from the UMD bay door to fit this in, it fit's fine, the door closes fine, no issue. Battery overall lasts about 12 hours and 19 minutes, I have timed this 4 times, on a running game, and believe me it's all you need it'll last very long, and charging takes just shy of 3 hours, using the original PSP AC outlet charger.
I would just not recommend the way that you have suggested to do it. Trim-pot (or trimmer potentiometer for those who don't know) is okay for use on various things PCB mounts etc, but when it comes to things like the PSP, then you might end up reducing the overall cycle lifespan of the battery, and possibly could end up short circuiting it thus increasing the Voltage back up to 4.4v. So for example when using it and it jumps back to 4.4v (the trim-pot failed) it messes up then it could cause a spark or short circuit the PSP rendering it useless.
It's understandable that many would like to use a battery lying around but if its higher in terms of Voltage, then the safer method would be to buy a cheap 3.7V 3000 mAh Li-Po battery and use the circuit board from an original battery or cheap knock off battery then solder with care and attach to the PSP.
The link to the battery you show, sorry but I hate those batteries, the new "Ribbon" type connector is not to my liking. I prefer the ones either with the solder taps for pos and neg protruding out, or with 2 12-14 single strand AWG wires attached (usually red/black), if you strip the top part of the battery (take the plastic connector off, then you will be breaking open the protective seal that house the actual battery inside, but as well you may end up causing a reaction (could end up smoking/fire) if not taking caution. If you decide to use a battery like that, then gently pry the top section off from the sides of where the plastic starts, this way you will end up revealing 2 internal connectors cut those, then take off the board/connector. Like I said do at your own will.
Otherwise good luck.
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