Today I experimented a little bit with the mass storage demo. I created the Dataflash.h and flash command header files based on the modified demo from https://github.com/JayconSystems/EPaper-Mass-Storage-Demo-LUFA and I got it to compile. Then I changed the pins and the flash id to fit my project. Flashed it and lo and behold — it enumerates! Unfortunately, I was unable to write anything to the flash — I suspect I need to provide the commands for this particular model of flash from its datasheet. More debugging to come.
I also didn't try to use the USBtoSerial to actually flash the ESP8266 — I will do that later.