AQ7280 OTDR battery does not seem to charge or hold a charge - what can I do to resolve this?

Be certain to use the PW823 model cable.
After you have confirmed that the operation of the battery itself is OK, then there is a problem with the OTDR main frame or the AC adapter.

And if the AQ7280 OTDR can operate without a battery, the circuit of the battery charge in the OTDR mainframe may be broken.  Repair of the main frame is necessary.

By process of elimination, you can decide if the issue is with the OTDR, the battery, or the AC adapter; the AC adapter polarity is very important.