Lord Business is clearly Finn's father, I've been thinking "gee, which character fits Finn" which for a long time I've thought it as either Emmett, or Vitruvius, which after watching the Lego Movie 2, I've felt it more to be Emmett as a better fit because he was the one Finn was playing with, and it was Emmett to break the cake, something Finn did, also it was Emmett who said what Phil said to Business, so if Emmett is Finn and Lord Business is (Finn's) father than it makes sense for Lord Business to be Emmett
which (if General Sweet Mayhem is Bianca) would explain why he didn't want to be involved with the duplo Aliens, because he knows his wife lives in the same star system and doesn't want to fight with her