@Xzayer
Probably to prevent potential rebellion. If a member of the previous royal line/lineage is still alive, it's possible they may be used (or may lead) a rebellion to reclaim the throne.
Since our emperor boi here is trying to rule all the continents, it'd be easy for a rebellion led by the "true heir/heiress" to occur in one of the countries when he's not looking.
Additionally, the people of that country may feel an ingrained sense of obligation to their previous royalty and assist in allowing that heir to reclaim the throne. If no heir exists, it'd be easier for a new emperor/ruler to take their place (kinda like how new male lions will kill all the cubs in a conquered pride that don't belong to them).
Basically, he's nipping all the buds from the get-go instead of letting an issue fester later when he's not paying attention.
That being said, I'm not sure if Pauliana mercy-killed the princess or just wounded her enough to where she's not a threat but can restart her life. Most likely the former, but I'd be interested in seeing her take a girl in a similar situation under her wing as a new soldier.