Suitable small/fast EEPROM for Atmega328
Sounds like you're just looking for nonvolatile storage, not necessarily an EEPROM.  Consider FRAM, like the FM25V01.  That particular chip stores 16Kb (larger sizes are available), has an expected endurance of one hundred trillion reads and writes, and has an SPI interface that can run at 40 MHz.  Your limiting factor here would probably be the ATMega328 speed.