The lamp sockets on the Mk IX range of sextants appear to be wired up to no particular pattern that I can discern. This would not have mattered when the sextants were built as they were intended to use filament lamps.
If you want to upgrade to LED lamps then you will need to know the lamp socket polarity. They can be of either type on the same sextant!
An easy method to ascertain the polarity is to use an ohmeter with one lead connected to the threaded ring at the bottom of the battery box, Connect the other lead to the case of each lamp in turn. If you see a low resistance this indicates the bottom of the box, the negative connection, is connected to the lamp body. I call this a Negative lamp socket. If you see no connection try connecting the lead to the smaller of the rings visible with the lamp removed. If you have a circuit then I call that a Positive lamp socket.
My tests, and reports I have received, indicate that an LED is an improvement to the bubble chamber position, to the extent that some apparently faulty assemblies work again. A lamp in the right handle does little apart from save current <10mA as opposed to 300mA. If both lamps are changed then a lightweight battery is a possibility. Using a lithium cell has definate advantages, these cells have an excellent power density, are unlikely to leak. I have never heard of, or seen one do so, and they perform very well in low temperatures. As Kalium batteries they were used in SARAH beacons years ago because of their low temperature performance and long shelf life. The Lithium CR2032 cells are usually cheaper than the standard "C" pattern.