I’m honstley not even sure what I should try next.
If you still want to continue trying to fix this portable, then i would suggest getting a battery caddy or something so you can easily switch batteries and remove them whenever you aren't actively trying to boot it. And/or add a on-off switch between batteries and the pms so you can disconnect them with ease at will. And btw you only need one battery to make it boot, adding more batteries only extends the runtime, so until you have fixed the portable, i would suggest only running one battery at a time.
As for the next steps in trouble shooting, i would isolate each system and test them separately, though primarily the pms as we are having power problems.
So, disconnecting the pms from everything else, and just have it alone with a battery, and maybe the status led, and check that there is no shorts, that it turns on properly and that all the voltages are correct, then add in one thing at a time, charging, fan, LCD, check that each one of them works before moving on to re adding the Wii itself. That way you will at least have a bit firmer ground and knowledge of what's working and where the problems might be.
For what might have killed the batteries, if there is not a direct short between b+ and gnd, then there might be a short on vsys to gnd? Vsys is always on, so it could have drained the batteries even when the pms is off. Otherwise it would be down to isolating the pms, and seeing if the problem still happens when nothing is connected to it, and if it is the pms that is broken, or if the problem goes away and that it is the wiring that was bad.
Edit: the status led is also connected to vsys, and is a potential culprit, so i would disconnect and rewire it too btw