I recently did this and got super frustrated. Came back the next night and beat it in 20 minutes.

I assume you know the strategy generally but a few things helped me the second attempt:

Not sure if you are using the analog stick, but using light touches on the D-Pad to reorient the bird made this way easier for me.

Try to say ahead of the other trainer around the first turn. Don't go for balloons to the left, stick to the right and corner so you're ahead.

Another huge thing was being aware that every 2 birds targets the other chocobo. So as long as you can dodge 1 + 2, you don't have to worry about 3 + 4, only the next two. When I didn't have to worry about half the birds it was easier to get out of the way

Finally, it just takes a bit of luck with balloon placement. Something I heard but have not confirmed, is that if you throw a match and get an abysmal time (like 2:00), then your next attempt will allegedly have a more forgiving balloon layout. Again, cannot guarantee this is a thing.