Thank you for the comments and input. I appreciate it.
I just got off the phone with the support guy at Davis. I'm afraid no definitive answers.
I asked whether it's OK to run a wireless SIM without battery and expect its supercap - if all is good - to keep the SIM powered through most the night. At first, the answer was 'yes' and then somewhat later in the conversation there seemed to be doubts as to whether there might be some or other software check built into the design to ensure that there's a battery installed... (Given all the confirmation I got here already, I think I'll go with his first hunch on this one though).
Then I asked whether the three sources (solar panel, supercap and battery) were all connected in parallel, to which he answered,"No, there's a distinct handover from the one source to the next, as and when required".
Lastly, I asked about the 12 sec on; 12 sec off transmissions and he said it didn't sound normal, but couldn't say for sure (he would need to ask one of their engineers). Anyhow, I checked up on the manual and the outside temperature updates at an interval of every 12 sec. It is the wind speed and wind direction that update every 2.5 sec. Since there was no air movement last night when I tested, I wonder if the absence of those sensor inputs may have affected the duty cycle of the transmitter in any way... ? (Just a wandering thought that's worth checking it out again tonight, I guess).
Also last night, I received my TP-Link router in the mail, flashed it with the Meteobridge firmware and have been updating the PWS Weather site successfully at 1 minute intervals ever since 2 am this morning (the SIM is running on a battery + solar at present). The readings seem to correlate with those of nearby stations.
So I guess I need to decide whether I should dump a circuit that behaves like this or not. Apparently Davis also offers a $60 RMA on the SIM board - so that opens up another option besides the $18 option of a PSU powered SIM.
(I realize that I come across as maybe stingy on this issue, but enjoying on a single income the luxury of a hobby such as this, in parallel to annual family health care expenses that ranges 20% to 30% of gross, one (sort of) has to be...
Thank you again for all your valuable input!)