Er, the way the Death Star's giant planet-destroying laser works is by superheating the core and causing the planet itself to go 'nova'. As far as I know the Dyson Sphere doesn't have a core, plus it's much larger. But also the Dyson Sphere doesn't have any effective weapons (though it does have a tractor beam). All of that is canon information, by the way.
Now in Broken Mirror? Since stats can't be made specifically for damage to a planet and such, the power of the Death Star's planet blaster would have to be high. And the Dyson Sphere doesn't fire the cutting beam but a few times, and it still doesn't do much damage to a Sovvie. So the Death Star would destroy it pretty fast in the game.
So canonically speaking, they would be at a stalemate. In Broken Mirror, I'd try to make it a stale mate personally (cause I like canon) but with the limited variables in the game, the Death Star would (unrealistically) win.