I did the macros for the dances on a new macro palette just so I could do this quest. It confused the hell out of me at first and I didn't get it down until like the third dance (I came in half way the first one and tried to watch people, but didn't catch on, so I went back to reading online for the next one, where I failed miserably with timing on targeting back and forth between the two).
Once you learn the timing of things (whenever they both stop attacking for a second and spout off their rant; target the "bad" imposter (the one not in the crowd) and use your dances against her, hitting the macro for the ability associated with mumor's chat message (see wiki for more info on that). Do that successfully (quick enough to count) and it'll give you a chat message saying success. Do that enough times (10 successes supposedly) and turn into the moogle to get event outfit. For me, I did 10 (I had them all in my chatlog so I know it was 10 exactly) and I didn't get the outfit. So I did it another time, with another 10 in my chatlog, and this time it gave me the outfit.
For me, as an older adult male, this quest just felt awkward and oddly embarrassing as I found myself questioning "what the hell am I doing with my life". I couldn't imagine explaining this quest to anyone I know and having them grasp it's importance or why a grown man would spend his time doing this... but there I was, fail dancing away until I got my trust.
I will say that if you don't get the trust, don't beat yourself up, this is easily the worst trust I've obtained so far, and outside of initial testing, I have no plans to call mumor out in the field for anything worthwhile, not even low-low-low level stuff where drain samba might be worth something.