It is possible but compatibility is really poor. You ended up using either PS2PSXe or POPS loader which emulates PS1 game rather than natively execute them. In PS1 native mode I/O chip (handles USB and Ethernet) inside your PS2 turns into PS1 CPU. So there is no way to get USB working on native mode. You're forced to use emulation method to keep USB port which again, very poor compatibility.
This applies to all PS2, Fat or Slim.