Honestly he and his faction only come into play after everything is settled. In the novel, he was mentioned AFTER everything was done so as to explain WHY the Duke doesn't absorb the rest of the Kingdom after they win, but in that, it's shown he is smart enough to just "take the throne and make peace" with them! Here in the Manga, he and his faction are mentioned early and to explain why not ALL the Kingdom has declared war on them! This is why he will be left in charge after his brother and father are dealt with! He didn't join their side, but just NOT aiding his father and brother and going neutral is good enough! He will suffer abit if they won for not getting involved, where for the Duke/Future King, not joining in worked in his favor, which will work in his favor when he takes over as King!
The reason not to take the whole Kingdom is it would be more troublesome, especially since the 2nd Prince and his faction are active and haven't been aggressive! What they do take is the Territories of the Nobles sided with the King and Prince! Tensions will be bad enough, and Allen knows better than to use many of his "Super Weapons" as that would cause ALOT of trouble down the line, they want to give the people stability after the War is finished, hence the 2nd Prince taking over the remains means they can "keep good relations, even IF they are the superior ones in this affair" remember that those in their kingdom might have realitives in that part of the Kingdom. One must know not to be greedy lest it blow in their face!
Don't forget the Empire still exists and it's STILL next door to Duke (well KING) Ramslett's territory, it didn't collapse just because the POS Imperial Prince was killed, It's an Empire, meaning there are MORE princes who could take his place! At best his death means they'll be more focused on "that" can of worms then the Split Kingdom! Likely WHY the Civil War can happen without their involvement. Their dealing with their "own restructing" and they can't even BLAME the Kingdom due to Allen having been careful, they don't know what a Plane is, so can't even GUESS where he went and by the time they DO find out about the Head of the Imperial Prince in the Kingdom's Capital, it will be too late to really do anything! With what they get at the end of the War. It's more then enough to make their fledgling Kingdom TOUGH ENOUGH to hold the Empire back when they get their own shit together!