I made standard batteries work
The internal batteries had pretty much totally died, so it was either change them or throw the thing away.
Everything was actually pretty easy until I had to fit the new battery pack back into the housing. In short, the standard AAAs would have fit had I not used so much solder. I'm not a very good solderer, so I ended up adding about 2mm to either end of the batteries, and because of that, they didn't really fit.
If I had been more careful (or had smaller solder to begin with, thus avoiding the massive pools of solder that I tinned the connections with), everything would have gone together just fine, even with standard AAAs.
As I said, just be careful not to use too much solder. Another option would have been to simply glue the connections onto the battery case and not solder the batteries at all. In hindsight, it probably would have been simple to glue the wires onto the top part of the case, and it would have been almost as easy to glue something onto the bottom that would have bridged the connection across the rear of the batteries. I think if I were to do it again, this is probably the approach that I'd take.