Sacking Constantinople does not guarantee the instant collapse of the ERE (it could mean so in the long run) nor that the Visigoths will held the city for longer. The way I see it is that the Goths will be eventually bought with their own kingdom in the Balkans or even Anatolia, like it happened in the West, so they will not migrate into the WRE. But it would mean a fatal blow for the ERE anyway, so you can imagine how an even more debilitated Byzantium will respond to the latter Hunnic invasion.
Meanwhile, in the West Stilico can held both the Rhine frontier and his own honour, so he's not executed. And with the Huns more concentrated in the East, maybe they don't bother to go West either.
As for how it could be done, there is an obvious answer: Treason.