What's the correct way to import FATFS? Getting error "Unknown type name 'DIR'; did you mean 'DDR'?"

It looks positive to me, as your compiling error is not stating it can’t see ff.h.

I think it might be due to FatFs not being configured ffconf.h, similar to when you first install LVGL and have to setup the config:
Go to your library folder for Arduino (Assuming you’re using Windows) ...\Documents\Arduino\libraries\FatFs\src

Rename or make a copy of ffconf_template.h to ffconf.h.

Now click build and let see if it compiles or what errors appear…