How can I determine the circuitry in this module?
There aren't a lot of parts.  The brute force method is to de-solder all the parts, then use a continuity tester to make a map of the connections.

A simpler way that will probably work is to use a ohmmeter in low voltage mode.  This is intended to not turn on silicon semiconductor junction.  In other words, a diode looks open regardless of polarity.  Not all ohmmeters have such a mode, but it's out there.

This won't help with resistors, but if all the resistors are large enough compared to a copper trace, then you can still distinguish a direct connection.

If you can't find a ohmmeter with a low voltage mode, you can rig up something yourself.  You're not actually trying to measure resistance, but determine whether the resistance is below a threshold.

One way you can do this is to start with a fixed current source, like 1 mA.  Clamp that with a diode so that the voltage doesn't exceed 700 mV or so.  Then look for the voltage dropping to maybe 100 mV or lower.  At that voltage, no semiconductor junction will conduct.  The effective resistance needs to be 100 m&Omega; or lower to trigger your continuity tester.