Is it possible to write data to a fat file system on SPI flash using an arduino?
The first and most important problem is not the filesystem. Currently on the board the USB port is connected to the internal MCU, which in turn is connected via SPI to the flash chip. The internal MCU is the master on the SPI bus. But the SPI bus does not support multiple masters on one bus. So just connecting the external MCU to the SPI lines as master would not work.
I see 2 possible ways to go here:

You can unsolder the flash chip from the board and add a fitting multiplexer chip, which can switch the SPI lines between the two masters. That effectively gives you only one master on the SPI bus at any time. This way would be rather hacky and difficult for a beginner. It might also be important to keep an eye on the capacitance of the bus, since it might run with rather high speed. A too high capacitance (for example through using a breadboard) might result in erroneous communication.
Then you could use one of the FAT libraries. I don't know them in detail, but I think you could also use them for a simple SPI flash chip. You just would need to implement the lower level access functions, that the FAT library needs.

Or you could try to use a USB host board with your external MCU and try to write data through the USB port. I don't have much experience with USB host boards, so I cannot really tell you, if and how exactly this is possible with the available hardware (the board will most likely implement an USB interface like a simple flash USB stick).


All in all I think, this would be rather hacky. Your board is not really made for changing the data through another microcontroller. Depending on what exactly you want to achieve, you might want to go to a different hardware setup.