There's no easy shortcuts on this one and can be difficult to hear, just repetition and use an instrument to sound out the chords to compare. It's a great way to train your ears and practice chords on your guitar/keyboard etc as well

btw, augmented triads are fully symmetrical so it's rather uncommon to talk about "inversions" with them, since an inverted augmented chord is just another augmented chord. at least as far as I've gotten, the inversion game doesn't include augmented chords for this reason.

This might help for diminished chords : If it sounds a bit wider and you can hear a M6 from bottom to top it is either 1st or 2nd. If you can hear the bottom note as the third of the chord its 1st. Maybe it also sounds a bit like a cheap 7th chord.

you need anchor points that you can recognize... focus on maj/min 3rds. If you can hear them, it will be very easy.