I've started getting a lot better at inversionist, and I put it down to a change in approach - I've done a search and a couple of these have been mentioned by Cuantas Vacas and others in different comments I think, but I thought I'd summarise what I've found most helpful. Bear in mind I'm only on level 19, but I've definitely had a step up in level success rate with this approach - and one which feels like I'm actually learning rather than fluking it or winning through trial-and-error with the help of an instrument or chord tool:

1) Write out chord intervals:
for each new chord type presented, I write out the expected intervals between each note of the chord including the first note repeated on top. So a major triad would be: M3, m3, P4 (intervals between 1, 3, 5, 1). I figure by constant reference to these intervals they'll start to become internalised.

2) Sing the notes you hear:
I try and sing the lowest note if I can recognise it, then try and sing the next notes in the chord. I refer to the interval pattern written down to try and figure out the inversion*. If the notes are too high to sing, try whistling them. If I can't recognise the bass note (I frequently can't either because of the instrument timbre or notes being closely spaced), then I start from the top note and work my way down. I try to sing all the notes of the chord in turn to firm up my choice - and if I'm still uncertain, I use one of the hints (traffic cone).

3) Always keep a finger over the c key:
Even if I get the correct answer, it's helpful to press the c key to get the spread chord played. I sing the notes as they're arpeggiated to check I didn't just fluke the answer. I always press the c key if I get it wrong.

*Bear in mind this requires already knowing intervals pretty well from departurer/lander/intervalis.

I'm hoping that over time instead of singing out loud I'll be able to sing the notes in my head and memorise the note intervals inside the chords without reference to them written down... Anyway I hope this helps somebody, and if not I apologise for the textual diarrhea!..