I have been reading this post for quite awhile and there has been a troubleshooting approach that is missing the most basic of test. First off this method will take about a month to get to the root cause. Maybe you have taken this approach, but I have not got that from any post. So here goes.
1) buy new battery and test voltage record value and temp of surrounding air at time of test. make sure battery settles for half an hour.
2) in the iss of concern, remove ALL sensor connections
3) install battery and test voltage and record. If sunlight is low or poor. check every hour till dark and record voltage.
4) Do this for the next week. Yes your console and internet data will not like it, but there is only one to find problem if there is a bad sensor, cable, that attaches to electronics module.
5) after a week, if you have lost 10% or more battery, then the issues are more than likely lack of sunlight, poor setting for the gathering of sunlight, or a bad solar cell or internal circuit leakage.
5A) remove the solar panel if you have a greater than 10% drain in battery and record voltage. run another week and check voltage. If voltage is stable, replace panel, still dropping, the super cap sounds like it is leaking. I see you replaced it. If you did not heat sink the leads and bend leads very carefully with needle nose pliers when installing, there is a good chance you damaged the cap.
6)So after step 4 you did not find a loss of voltage, then its time to find out which sensor it is. 7) Plug in anemometer for a week. check voltage, any loss greater than 10% yes, this could be a bad sensor. if not disconnect and plug in the tipping bucket.
7) keep this up till you find the sensor that is drawing all the current from the battery.
now plug in all the sensors that did not have a large effect on the battery, check battery voltage and run another week. if minimal lose of voltage is measured, then you found your sensor.
Even though we live in an electronic world, all devices use current to function. If ohms law or any calculation you would use to calculated current draw for an electronic sensor, transistor, ic, etc, this is the only way to truly know what the calculated use would be. Another check is the environment. Do you live in high humidity area. This can cause leakage and with dirty air or even salty ocean air to create a resistive circuit to form and now you have a load which will use current and drain battery.
just my 2 bits. take it or leave it. hope this helps.