Try leaving the battery unplugged while having it connected to a power source, I am actually going to go grab mine that's been dead for at least a month now and see if I get the same symptoms
You could try plugging it in while the battery is disconnected, there are a lot of users saying the batteries die die in them faster than CMOS batteries which is insane to me since the regular PSP batteries are like tanks
I would recommend looking on a site called tindie: adapter-flashcatusb-xport-360-clip-tsop-48-nand/
I was able to find a 360 clip to flashcatusb xport adapter and for the 360 clip not staying on I have seen people use books, binder clips, glasses of water and even rubber bands to keep it in...
I was wondering if you could supply a parts file and their locations for an individual manufacturing run, I couldn't find any mentions of parts in the GitHub