I have now tried different approaches to save power with my setup, but unfortunately nothing really worked.

I have turned every unnecessary module off via the PMIC. With this setting the M5Stick+ENV Hat were five days in operation. After this test I thought that maybe the ENV Hat draws to much current. So I repeated the test without the Hat. This gave me one day more run time.

The deep-sleep mode that I used is actually called light-sleep. In this mode the RAM still has power (see https://lastminuteengineers.com/esp32-sleep-modes-power-consumption/).

So I tried the the real deep-sleep mode. Unfortunately in that case to Micropython scripts starts from the beginning. So I changed the code a little bit. I made it to connect, measure and send the data to the RasPi every hour. But the connecting costs to much battery power and this setup worked only for one and a half days.

Maybe I could save the data in the flash and then connect once a day like planned, but I am to lazy and do not have enough motivation to finish this project.