When we are trying to understand a difficult passage like 1 Samuel 15, it is really important to get a good idea of where the passage fits into the big picture. And that's particularly true with this one. In particular, we need to understand why Israel was important.
At this point in the story, Israel are God's chosen people. But they weren't chosen so God could bless them and curse everyone else. They were chosen to be God's conduit of blessing to the whole world. As God's original promise to Abraham says:
all peoples on earth
will be blessed through you.
Genesis 12:3b, NIV
Israel was God's chosen conduit of blessing to the whole world. Amalek had actually had a chance to be there as well, being descended from Esau. But Esau had renounced his blessing, trading it in for a bowl of soup, and Amalek continued in that. They had decided that they would oppose the very means that God had chosen to bless them and every other nation, and by the time we reach 1 Samuel 15, they have been consistently opposing it for hundreds of years and show no sign of letting up.
In his book Violence, Hospitality and the Cross: Reappropriating the Atonement Tradition, theologian Hans Boersma points out that hospitality requires the potential for violence. Suppose that Britain welcomes a refugee from Burma. In Burma, they are being hunted by the authorities because of their statements about human rights violations, or something like that. If Britain really welcomes them, part of that is being willing to resist the Burmese government sending agents over here to kill them, and resisting in a violent way if necessary. Part of hospitality is willingness to protect the people you are being hospitable towards.
In the same way, God is determined to bless the world, and at the stage of 1 Samuel 15, the way he has decided to bless the world is through Israel shining as a light for him among the nations. As it turns out, they're rubbish at that, but that's a different story. Even so, we still get people like Ruth and like the Gibeonites coming in from outside Israel to experience some of God's blessing to the world through Israel. And so part of what it means for God to bless the world is for God to protect Israel, his pipeline for blessing to the world.