Okay finally got this figured out. The trick is to go to the First Order area not wearing Resistance clothing (wear a standard Batuu outfit), and talk up one of the First Order sims wearing an officer's uniform (except Lt Agnon, who even with a great friend score never gives the chat option) until your friend score is high enough for the options to "Seduce/Bribe/Fight for Uniform" appear. If you succeed at any of those three, you get both the First Order Uniform and ID Card steps of the mission checked off, and both items appear in your inventory. The First Order ID isn't usable (but I suppose you give it to anyone who checks your ID when asked to keep from being escorted to jail), but the First Order Uniform can be worn through a dialogue in your inventory itself, which is necessary for the last step of the mission.
