Electrical – Is it possible, and if so, how difficult would it be, to replace the volatile memory in Gameboy games with non-volatile memory

What you need is a technique called wear leveling. It doesn't write your data every time at the same location in the EEPROM, but uses some algorithm to use different locations. I've read about complex wear leveling algorithms, but I wouldn't know why the following simple method wouldn't work.

Add to your data a 24-bit counter, so that your data block is for instance 8 bytes long. Pages on a 24AA64 are 32 bytes long, so a 64kb EEPROM holds 256 pages. From the datasheet:

"When doing a write of less than 32 bytes the data in the rest of the page is refreshed along with the data bytes being written. This will force the entire page to endure a write cycle, for this reason endurance is specified per page."

so it doesn't make sense to use data blocks smaller than a 32 bytes page.

Look at the counter of the first page. If it's zero you used the maximum number of write cycles for that page, so you move on to the next page and check that counter. Repeat until you find a counter > zero. That's the page you're currently using. Microchip's EEPROMs have a 1 million cycles endurance, which you can increase to 256 million with the given example of maximum 32 bytes per block in a 64kb EEPROM. That should be enough to outlast your product: 40 years if you write once every 5 seconds(!).

You'll want to initialize your EEPROM on first use. How do you know when that is. Use the last page to write a unique signature upon initialization. Check at each power-up if the signature is there. If it isn't the device has to be initialized. You can preset the counter in each page with 0xF4240 (for 1 million) or clear everything to 0xFF and write the 0xF4240 when you first use the page.
Initializing an EEPROM is needed because sometimes a certain pattern is written to it in the production/test process.

edit
The wear leveling should solve your problems, but I still want to comment on the capacitor solution. You say the board is rather power-hungry, but maybe you can isolate the microcontroller/EEPROM's power from the rest of the board with a diode. So you'll probably need only a few mA when main power is gone. The 24AA64 writes a page in less than 5ms, then at 10mA and an allowable voltage drop of 100mV you'll need

\$ C = \dfrac{I \cdot t}{\Delta V} = \dfrac{10mA \cdot 5ms}{100mV} = 500\mu F \$

Easy with a small supercap.

further reading
datasheet 24AA64
EEPROM Endurance Tutorial

Probably not, but then writing the simple code for this in a microcontroller is really easy. EEPROM chips don't work this way because there is more to a EEPROM than just storing and reading back a string of 500 bits. Some SPI EEPROMs come close to what you want once you get going, but there are still issues of pages, opcodes, and the like.

500 bits is only 62.5 bytes, so any micro with at least 64 bytes of EEPROM built in can easily do this. That is a small amount of EEPROM for a micro. This also allows the button debouncing to be done in firmware.

Also, 500 bits by hand? Seriously? The chance of getting those all right is pretty much zero.