Well, since you seem to have some sort of Internet access, then you probably already know that a 24LC256 serial EEPROM has a storage capability of 256K-bits, whereas a 24C16 has a storage capability of 8K-bits, since you probably already downloaded the datasheets and compared the differences between the two devices.
You can't get a 24xx256 serial eeprom? This chip has been out for literally decades. What prevents you from acquiring this device?