There are two types of Battle Score Bonuses that you can earn in Allegiance Battles: the Advantage KO Bonus and the Pair Up KO Bonus. Keep in mind that you will lose points when your allies are defeated, so make sure you don't let anyone on your team fall in battle.
This is a +10 score bonus earned when you defeat a foe using an ally with weapon-triangle advantage, or by defeating a colorless foe with a colorless ally.
This is a +5 score bonus earned when you defeat a foe using a paired-up Hero. If you include Legendary Heroes with Pair Up on your team and get everyone paired up, you can guarantee that you'll receive this bonus every time you defeat a foe.
Scoring an Advantage KO Bonus with a paired-up ally will grant you the Pair Up KO Bonus as well, for a total +15 bonus to your score.