Eugene's set-up for the story makes it unlikely the son was in the building at all. He arrived with a takeaway meal for the two of them (as he apparently often did) to find Fyron and Xykon mid-standoff, and as he entered he was calling his master's name, not any others.
This just after a montage of scenes throughout Eugene's time with Fyron, too, with narration talking of the father-son relationship they had. If Fyron already HAD a son, that would rather complicate matters, or else they would all be one happy family. I can't believe that Eugene and Fyron could be that close and yet the son's death doesn't warrant a mention.
There was no hunting down. Xykon took a tour of Fyron's museum, saw a crown that he thought looked totally badass, and returned that night to steal it. He didn't even necessarily intend to kill its owner; he just wanted the crown.