External Flash Memory File System
I don't think you need a file system: that is usually needed when you want to manage many files. Your weather data log sounds like a single file.

What you seem to need is to access EEPROM as a stream, exactly like you would for an open file. There are stream wrappers around some EEPROM libraries, e.g. this one for ESP. I suggest you check it out and see if you can adapt it to the EEPROM library you use.