After about 3 weeks (I don't know exactly by I documented the dates here somewhere) I started with a full battery and solar power. The last few days were sunny but still yesterday to power for the microcontroller went out. It also produced weird sensor values the days before so I guess the power supply was instable. Two days ago the microcontroller also was in a weird state were the power led was very slightly on and the wifi led on full power for several minuets. This means there are also software bug/edge cases I have to take care of.
I have reviewed FLASH, EEPROM and FRAM a bit more and FRAM would be ideal but as they are quite pricey I will go with an extern EEPROM for now. The one I am going to use (25AA010A) is rated for 1M write cycles, so if I use the same cells over and over again (which I will not do) this means 24 writes per day. 1M/24 means 41,666 which are about 114 years. So this will probably outlive me. Even if I first clear the EEPROM this means 48 writes per day so only 57 years which is still good enough.