So I figured it out, and it's not straight forward at all. Once you complete the achievement, log on to a character who has completed Chapter 9 of KOTET, not KOTFE, so has completed the entire Eternal Empire storyline. Go to the Legacy window, and scroll down to the Feats of Strength achievement. Then right-click on the item shown and that will finally trigger the Alliance Alert. Darth Hexid will show up in the Force Enclave on Odessen, and you'll be able to talk to her. Once you do, she's available as a companion immediately.