Here is a complete working version of FAT16 on the USB side. Think of this project as a driver to fool the operating system to believe this is a USB you can put data onto. If you've wanted to hide secret messages on the pico or use this project as a base for other projects, be my guest.
First, install the cross-compiler sudo apt install cmake gcc-arm-none-eabi libnewlib-arm-none-eabi build-essential libstdc++-arm-none-eabi-newlib. Then run git submodule update --init, cd pico-sdk, and another git submodule update --init. Then go back to the root directory and run these commands or compile.bash:
mkdir build && cd build
cmake ..
make
TODO
While the pseudo-USB reports it is 128mb big, in reality you can only write about 1mb to the pico in total. This is because how your computer determines if a FAT filesystem is FAT12, FAT16, or FAT32 is determined by the amount of clusters that the data section can hold (Microsoft's FAT Whitepaper), and the 2mb of flash memory that pico has will not cut it.