Sunset's not used to fighting such bizzare monsters, considering his time period, but it can't be any worse than a gigantic villain with a gatling gun, can it? Regigigas is slow. Not only that, but his damage is terrible for 5 turns. That means that Sunset has all the time in the world to do whatever he wants. Hurricane shot should outright kill, if not seriously maim, his opponent, and while usually the 3 turns it takes to use make it useless, here it's exactly what he needs. Of course, Regigigas's paltry attacks are nothing to worry about, and there's nothing it can do to avoid a Hurricane Shot. Speaking of avoiding attacks, even if Regigigas manages to pile on the hurt, there's a good chance Sunset will evade at least some of it anyway! With all this in mind, Sunset definitely has the edge over his Poke opponent.
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Regigigas isn't much of a gunslinger, but it doesn't matter. Sunset has no invulnerability to Confuse Ray, which will leave him helpless, and, more importantly, cause him to lose the advantage given by Slow Start. Regigigas's attacks, while weak, will still cut through Sunset's terrible defenses like a knife through butter, and his own natural tankiness can absorb punishment while he slowly whittles away at Sunset, should the cowboy break through Confuse. A few Mega Punches here and there, and he can use Giga Impact for a finisher and end the fight before Sunset can get off Hurricane Shot. And should the battle happen to last past 5 turns, Slow Start works to his advantage, and he can easily destroy the frail dueller. It might not be short, it might not be pretty, but Regigigas can win this fight, and with any luck, he will.