Lowering the profile of the puck to make it fit a tight spot, how to power?

The big pad on the back is actually battery negative, not positive (because the 'shell' on the battery is +). You do have the two labelled GND and 3V pins here as well that you could use: http://www.espruino.com/Puck.js#pinout

I'd consider sticking 90 degree pin strip into the 0.1" holes, then you could use simple 0.1" socket (like the 'dupont jumper wires') to connect to it, without making the Puck much thicker.