Thank you, rdsman. Those are very useful suggestions which I'll definitely try out next.
It's worth noting that I have replaced the anemometer as well, since the bearing on the one that came with it was worn. (It was the old type and so I believe I couldn't just replace the bearing itself). So if excessive draining would be sensor related, I would expect it to be coming from the rain and/or humidity/temperature sensors rather.
Dalecoy, yes, when I got the station I measured the battery and it was sitting around 2.23 V. At that time I incorrectly assumed it to be a rechargeable battery and so I did not think much of re-inserting it with the hope that it should charge up again.
So at first, I simply fired it up with the supercap and battery that came with it. Initially it would keep going through most of the night, but would go dead +/- 2 or 3 hours around sunrise. However, that duration increased over the following days to the point that the SIM was dead by around 8 pm already. That's when I pulled out the battery and tested it - it measured 0.8V only. (And that's when I also discovered that these weren't rechargeables...)
While I was waiting on the replacement battery to arrive, I ran the station without its SIM battery; with the old supercap + new solar panel. As soon as the sun set, the SIM would be dead also. There were no signs of cap leakage or any other sign on the circuit board for that matter, that made me suspicious. But I went ahead to order and replace the supercap nonetheless. I replaced it with an exact replica: Nesscap 2.7V 10F.
This is where I'm at now - trying to make sure all is well before I mount it - inbetween 7" of rain that came down last week, etc...
I guess I also have the option to run DC up the pole. But I read elsewhere on this forum that some wireless units would act up and give erroneous readings if the SIM gets powered externally...