Hmm.. interesting question. I don't know if it's even possible to program a AT25320 with an ISP.
The AT25320 is an SPI device, and while an ISP uses SPI to program a microcontroller, the signals the programmer generates are tailored to the uC in question. To program an AT25320, you'd need the READ and WRITE commands listed in its datasheet.
If you want a piece of dedicated hardware to do the job, the Bus Pirate would be more the tool you need: http://www.adafruit.com/products/237
. Its whole purpose is to translate commands from your computer into messages in various protocols (SPI among them).
You don't really need that though.. any Arduino can do the job. The ATmega328 has built-in hardware to support SPI communication, so it can read and write data to the AT25320 all by itself. A Trinket would be able to handle the communication, but might not be able to hold all the information you want to write into the EEPROM.
When you void a product warranty, you give up your right to sue the manufacturer if something goes wrong and accept full responsibility for whatever happens next. And then you truly own the product.