Thanks for the extra data buy after looking for several readings that only alter by 1 bit in the data (not CRC) I cannot find what I'm looking for. It could take weeks to collect enough data to get the value I need and a long time sorting the data to find them.
Only idea I can think of is a little drastic if your not into soldering and that's to de-solder the temperature sensor so the temperature will never change and only humidity values will alter.
As to the battery indicator bit, yes this probably could be wrong. Does the base station have several degrees of battery power indication (like bars on a mobile phone battery indicator) or is it a simple good/bad indication? Powering the sensor from a bench PSU would allow accurate voltage adjustments and the battery indicator on the base station could be noted as the power is slowly decreased and the data captured. This would help narrow down the battery bit(s).