Just bought a secondhand PSP 1003 for my ex-wife

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My eldest daughter and I have the 2003 versions so different battery.
When I plug it in with a the battery in situ, I get the orange light, but then the system shuts down after a couple of minutes. This happens even if I am using it via the mains adaptor.
Battery is showing full number of bars. When I unplug the charger the green light comes on for a second and then everything shuts down again.
System works fine with battery removed. If I put the battery in whilst on mains I get the orange light and it shuts down like before.
Is this a faulty battery telling me lies? I have ordered a 3600 Ni-MH one to replace it.
I hope it is not the PSP as I only bought it because it was mint condition. Shop has plenty of second hand units and are willing to swap if there a problem and it has a years worth of guarantee.
If I have to swap does the front detach I don't want one with all scratches over it.
Regards
LFM999