Oh man. This mini-game is a big nightmare, so I'm afraid I won't be much use to you if I attempted to do it for you. I don't think I've ever actually managed it.
I've found this for you where someone managed to do it, if you'd like to emulate their movements as best as you can. It looks like they are mostly keeping to the right to avoid the birds, and only dipping in to get a few balloons when the coast is clearer.
That's unless someone else here has managed to beat this.
Sadly when I did a video playthrough of FFX, I did a video on the Remiem Temple chocobo races, but not the Catcher Chocobo. I've succeeded in doing this three times, maybe? So I wouldn't go nuts on trying to stay ahead of the opponent. Just worry about getting the balloons and dodging birds. I find that staying to the left side once you get to the bottom sort of gets you out the birds view a little and you have to dodge less. And you sometimes see the birds get to your plane and you can tell they're going for you, so pick a direction and start moving that way, which it usually will move that direction twice, and you can usually dodge it. Keep your composure. When you get near the Arena, your angle changes and the birds are basically impossible to dodge because of how they approach you, so you'll need some luck there. Again, stay to the left and even go down that mini-path there if you want to try to avoid getting hit. It's hard to describe, but you sort of get a feel for it and learn to maneuver well. It's not totally iron clad, but it's a solid strategy. Sometimes you do all the right things and the birds just find you and you can't get away. But doing it this way, you'll eventually put in a good enough run to get you over the top. I usually work it for roughly a half hour before I land it. Unfortunately, luck is a factor, so just try to put up a good run and if luck finds you and you stick with the plan, it'll eventually land you your Sun Sigil.