Update: I ended up using Coilcraft DA2320 in place of the blown transformer. The pinout of that transformer is very different, so I made a tiny adapter PCB that sits between the BMS board and the transformer.
Another possibility is P0926NL from Pulse Electronics, although I haven't tried it.
DA2320 is not a perfect (or even a close) match, and one of the rails now reads 11.5V instead of the 8V indicated on the silkscreen. That doesn't seem to matter, as there is a voltage regulator to generate the needed 3.3V, and current consumption is pretty low to worry about efficiency.
I've charged / discharged the battery a few times after replacing the transformer and so far all is good.
Hope someone will find this info helpful.