I2C EEPROM programmer

Each of the binary files is 16kiB, including the file I intend to use.

If you are going to hand enter the data like the example provided, you can use multiple entered arrays and just manage the eeprom address where the storage takes place. There will be memory issues if your array blocks get too large.

Alternatively, you can put all the arrays in PROGMEM to store the data in flash (program) memory instead of SRAM, but that adds another level of complication to read them. Using multidimension Arrays with Progmem - Programming Questions - Arduino Forum

If you indeed have 1000 unique lines of 16 byte data fields, you may want to develop a tool to get the data from the computer into the Arduino line by line. Google will be your friend here, and you will probably explore "Processing" and "GoBetweeno". If you are entering new territory here(like I am) it may be faster to brute force and hand code the solution.

If the data is on an SD card, there may be examples of how to read line by line into an Arduino in the SdFat library examples.

At this point, you don't have an eeprom problem, you have a data file management and pc/arduino problem. I'm not very familiar with this area so can't be of much help. Google is your friend. Good luck.

The background is this: I purchased a radio for my car. The car is a 1995 model, but the radio is a 2014 OEM from another manufacturer. As it turns out, this radio was never commissioned, so instead of asking me for a passcode, it requests a diagnostic tool.

I don't understand this. If the radio is not stolen, I would think that the manufacture should be able to supply a passcode based on the serial number. What does "never commissioned" mean?