I've had the most success with the method you have already described, listening to the outer voices and trying to determine if I hear a Perfect 5th (Root), Minor 6th (1st Inv.), or Major 6th (2nd Inv.). Which is easier said than done! My ears aren't great at hearing the differences between those intervals, so I also listen out for the chord quality on a scale of bright-to-dark. For example, Root feels the brightest, 2nd Inv. feels less bright/more complex, 1st Inv. feels the darkest. *This bright-to-dark scale changes when you're dealing with Minor Triads in later levels.

Thanks for your comment, there are some things that i didn't realize in inversion chords (like the quality of chords, bright to dark) so i'm gonna try this approach and keep trying to listen the intervals for the inv. Thank you !

ToneGym is missing an allocated area for tips for each game hence I'm repeating my advice here, apologies for those who have to see the same post copy and pasted.

First of all, the game should have had the rating "advanced"

1) So since this game is advanced I encourage you or I would recommend using a musical instrument or even an app on the phone like a virtual piano, don't fret and don't feel guilt about it, over time you will be able to drop this aid (promised)
2) 2) use your instrument to detect the lowest note you hear (the bass)
3) 3) observe the name of the given chord, play the chord for yourself if needed and observe its structure.
4) 4) determine which degree of the chord does the bass represents
5) 5) over time when it becomes easier to hear and separate the bass note, try to detect the next stage's bass by comparing it to the previous bass you heard, eventually that would enable you to forego the use of a musical instrument. But there's no rush.
6) 6) this game requires you to sweat, decide on how much time you're willing to invest in it and then stick to the training religiously it doesn't matter if you finish a level or not but stick to the workout on a dedicated DAILY religious like routine
7) please report back on how do you do

Thanks Yooval, i need to go rn but i'm gonna try your technic and report it back to you by dm

This post has been very helpful, my thanks to the contributors!

Thanks for repaeting @yooval mann Always welcome your knowleg

Thanks Cedric please do whenever you can.

You're welcome Lucas thank you.