Hmm. Sight singing and dictation was something that I never had to practice, and I really don't know how to teach it. I need to learn, though, since I plan on teaching it one day.
Here's a suggestion. To my ears, a chord in any inversion is recognizable. So, try playing a chord in all of its inversions, and listen for the similarities
in the chords, so that you can recognize that chord no matter which inversion it appears in. After you can recognize those similarities, it should be a lot easier to sort out the inversion of the chord.