Isn't the point (and I could be wrong) with a Dyson sphere that it should have a spectrum that approaches that of a perfect blackbody radiator? Or would any radiation that sneaks past the swarm drown out the blackbody spectrum?

> Isn't the point (and I could be wrong) with a Dyson sphere that it should have a spectrum that approaches that of a perfect blackbody radiator?

Pretty much. The lower the temperature of that blackbody radiator, the higher energy you can extract via temperature difference. Until you reach the background radiation of the universe, which could be considered the lowest possible temperature of a thing that still extracts energy.

> Or would any radiation that sneaks past the swarm drown out the blackbody spectrum?

If that's the case, you can improve your sphere and capture that extra energy.