How to read/write variables persistenly on Arduino Due (no EEPROM/shield)?
I really hope that somebody knows differently, but I'm fairly sure that the answer is that you can't directly save runtime data back to the Arduino Due board.  I've looked into this myself and have only found answers saying "wire up an external EEPROM" or similar.  Whilst it seems that it should be technically possible to write into unused (or pre-reserved) areas of the program memory, I suspect that the CPU has to be put into a special mode to do this, which is very probably only possible from the bootloader when flashing a new program to it.

I've chosen to go with FRAM chips for data storage. Link to mine.  They are difficult to wire up - the only ones I found were in surface mount packages (although some hobby retailers sell them mounted to little boards with header pins), but they are very cheap, very durable (you can write to them a great number of times before they wear out), operate at high speeds and you can read/write single bytes from/to them.  Other storage chips often require you to write whole "pages" of data at a time which can mean reading in the existing data, modifying it, then writing it back.

Additional comments:
In your original question, you state that the contents of your PROGMEM located array must be restored on restarts.  I don't think this is the case.  That would suggest that the contents are stored elsewhere and code is in place to overwrite them, however it could be the case that they are copied into working RAM, which would mean that the PROGMEM macro is not actually doing anything on the Due - which may well be the case.

Modify your code to change one of the values of the array (within the bounds of the array!) and then print its value out again after changing it.  If the value printed has changed then this means that PROGMEM is not actually doing what it is supposed to do on the Due, which would be a real shame.