So one of my batteries for the new project died. Upon testing some hardware, it turned out one of the batteries had incredibly low voltage. The battery wouldn’t charge properly; it seemed one of the cells had been damaged. After removing the top casing and topping up the damaged cell, there was still no real change in condition.
Stabbing around in the cells revealed that there was a high resistance between the outside terminal and the cells inside, and the battery could be charged in some manner by attaching to the cell directly, bypassing the outside terminals.
So then this happened:

We cut a hole in the battery next to the original terminal to get easier access to the plates inside. The plan is to create a new terminal connected directly to the lead plates inside.
It seems that a thin lead connection from the plates to the original terminal had broken; this is probably a safety measure - the connection is possibly intended to fail in the event of a short circuit. It’s not obvious whether this connection failed due to mechanical causes or due to an accidental short circuit of the battery.
Hopefully this all works out and I get a functional battery again. Otherwise the project may be delayed for some time.
Don’t try cutting up batteries at home. It’s dangerous. I take no responsibility for what you do. Cause I’m cool like that.
Update: We attemtped to charge the battery with the new terminal setup, and it went all right, though the battery started hissing and popping - we were overcharging it, leading to release of hydrogen from the battery. We topped up the cells with a little distilled water to account for this.
To replace the connection inside the battery that had broken, we decided to just solder a wire from the lead busbar inside the battery to the terminal outside. Unfortunately our standard soldering iron wasn’t hot enough, so we decided to try heating up a piece of brass rod on our gas cooktop.

Sadly I didn’t get an action shot. In any case, it didn’t work well, so we borrowed a hotter iron and got it done.

To make the battery usable again, we covered the hole with a small piece of plastic and some sealant.

The battery seems to be working fine now, charging and discharging as expected. This cost nothing to do, saving me a whole $30 AUD for a whole new battery. In summary: Awesome.