Tutorial: visually teaches the nuances of this star.
Tips
To fit in the starting triple jump more consistently, you can hold neutral briefly while airborne during the buffered kick. Doing the whole triple jump with full stick magnitude requires very good early-frame jumps and short A presses.
Make sure to long-jump early after entering the slide to land further back and be able to build up speed on the slope before sliding off.
While falling, hold up-right to turn, switch to up to clear the slide while avoiding turning too far right to bonk the wall, then switch to down-right/right to drift closer to the bottom of the slide.
Switching to holding right with good timing during the slide yields a consistent path across the bridge.
Hold up-left as the slide ends to steer towards the door and so shorten the distance of the (following) moonwalk, meaning you'll have enough residual speed to reach the door.
Hold up-right after the rollout to moonwalk in the correct direction (you must have passed the finish line facing backwards).
Compared to the beginner strat, we also slide across the gap by the far-left corner of the ice bridge, triggering an instant moonwalk.
Holding (full-extent) down-right as soon as the moonwalk starts is a setup for hitting the door, but it's very sensitive to angle of approach and minute changes in stick direction (holding closer to right will make Mario turn more, and closer to down will make him turn less).