But... what I do have is a Bus Pirate!
Here, I've connected the 3.3V rail of the Bus Pirate to the battery terminals on the T39m. The Bus Pirate probably won't be able to sustain the current draw for an actual phone call, but it can certainly supply enough for the phone to boot:
Success! And a full strength signal too (that's old-school GSM frontend design for you - none of this "you're holding it wrong" rubbish). Aside from the zeroed clock and the dead SIM, the rest of the phone works fine and I was able to extract the data I wanted.