Why might the omniscient narrator choose not to give a lot of details about the fight between Rikki-tikki and Nagaina in the hole?
Great question! This was probably done to heighten the tension and drama of the scene.
As readers, we were treated to all of the details of Rikki-tikki's fight with Nag; these revealed more about the mongoose's character, especially his courage and tenacity. But then the final details of his fight with Nagaina are withheld from us.
We do get to read about how Nagaina slips into the hole, with Rikki-tikki following close by, and we're told that the hole is "dark," that the little mongoose "never knew when it might open out" and allow the huge snake to turn and attack him. But then the narration returns above ground to focus on the now-still grass by the entrance to the hole and on Darzee singing his mournful song. The next thing we know:
Rikki-tikki, covered with dirt, dragged himself out of the hole leg by leg, licking his whiskers. Darzee stopped with a little shout. Rikki-tikki shook some of the dust out of his fur and sneezed. "It is all over,"' he said. "The widow will never come out again."
As you can see, the narrator probably omitted the details of the death blow that Rikki-tikki dealt to Nagaina in the hole because, if we can only see Darzee singing a pessimistic song while there seem to be only silence and stillness below the ground, then as readers, our curiosity is extremely piqued: we're on edge, tense as can be, waiting to find out what's happening and if Darzee's song of mourning is appropriate or not.
This story is a theatrical one, and very entertaining. It's a lot like an action film. Imagine watching the climax of an action film. The hero and the villain are fighting on top of a train, and then the train disappears into a tunnel. We can't see anything. It's so exciting! What will happen? Then the train emerges into the open air, and we see the hero standing victorious. Relief and joy fill us. The same thing is going on here in "Rikki-Tikki-Tavi." By essentially blinding us to the final action scene, the narrator is keeping us even more on edge than we would be if we could actually read the visual details of Rikki-tikki ending Nagaina's life.