ATmega48PA power consumption during EEPROM write
The complete datasheet for the ATMega48 has more details on power consumption, but nothing specific to EEPROM write. It's not unusual that writing to EEPROM would take a lot more current than regular operation, however.

Instead of powering your AVR via a large resistor, you should use a linear regulator so its input voltage doesn't depend directly on the current it consumes.