I have no explanation for your rain spikes, let us hope Ecowitt supplies some information and please share that if they do. This reply is more about the battery/capacitor.
I do have some anecdotal observations on the charging/discharging of the capacitor, and battery status when not plugged in. I would also be interested in others experience.
1. I notice the capacitor depletes faster when there is a rain event (this might make sense).
2. I notice that right around 2.5V on the capacitor, the battery V starts to fall a small amount, like 3.28 to 3.26 , this is with Lithiums. This can go lower depending on how long the capacitor is held low and dropping due to lack of sun. Normally in my environment the battery bounces back to 3.28 V once the cap gets charged up again via solar. This is of course, not battery charging, but probably the normal bounce back when the load is removed from the batteries. Lithium batteries have a much flatter discharge curve than to alkalines.
3. Although you can get some capacitor charging at like 50 W/m^2 , really takes a bit more than that, and for a long period.
4. My capacitors on both units top out at ~5.3V when charged.
Now , I looked at your Solar chart, in my opinion, that is not enough to get the supercapacitor to charge in a timely manner (meaning I doubt you are getting to a full capacitor charge each day, or any as far as I can tell). I have a short window for charging thanks to trees, but my peak solar is usually around 350 W/m^2 (this time of year with a low sun angle), and that , is not enough to get the caps to full charge, unless I have a few days like that.
Is your unit shaded? for a large portion of the day?