DFPlayer for Writing to SD Card
Unfortunately the DFPlayer cant write that data to the SD card. The Arduino has some non volatile memory of its own built in though. Theres 1K of EEPROM for the UNO. Notes on how to use it here: https://www.arduino.cc/en/Reference/EEPROM
If you can get what you need to done in 1K and its not important to your project that you be able to unplug the SD card and put it in another reader, then perhaps thats a nice, simple solution...