I completed the Alien Derelict with no problems. 3/3 levels cleared.
I moved on to the Robot Derelict, only for completion of the first level to result in 4/3 levels cleared and that mission terminating.
At which point I also discovered that my entire team now has Scout equipment like my Commander instead of their actual equipment.
(Closing and reopening does nothing to fix this.)
Going to see if I can't back up the save file, start a fresh one with the relevant classes, & copy their equipment over to correct the original.
Seems like that should work, though I'm not sure whether I can fix the level completion bug in a similar fashion.