The final test shows the 2011-F pack may power an LED blinky, but not much else:
Although the total capacity is still about 1.3 A·h for the two best batteries, the camera says the weakest two are dead after a few photos.
For reference while resoldering, the joints at the negative terminals:
And the protection PCB on the positive end:
Unsolder the strap in the middle and the B+ positive connection on the right side to remove the cells.
If cameras used bare cells, rather than glued-shut “proprietary” packs with super-secret unique ID ROMs, they’d be easier to keep running. My Sony DSC-H5 has other problems, but NiMH AA cells are easy to find.