Whenever you see a shotgun in a story, it's a symbol. (We don't know if that's actually true or not, but it's probably a pretty good bet.)
And it's definitely true for "Rikki-Tikki-Tavi." Both the shotgun and the stick represent man's control over nature and his ability to bend nature to his will. And by bend nature, we mean kill it. See, Kipling's protagonists are generally go-getters. They prefer action over something namby-pamby like science, politics, or understanding. They get the job done Schwarzenegger-style, and that's about that.
Teddy's father is no different. Sure, he nurses Rikki-tikki back to health, but he understands mongooses to be useful critters that will be friendly when fed (12). In other words, the mongoose is not a threat—unlike snakes. The father sees Karait as threatening Teddy, so he comes to beat him (the snake, obvs) to death with a stick (36). When the father hears the scuffle in the bathroom, he blows Nag away with the shotgun (61). (Too bad Rikki-tikki got there first.)
Rather than understand the threat or try to rearrange his garden to minimize it, he simply uses his power as a human to remove the threat, permanently. Whether you see this as a positive or negative aspect in the story will depend on your reading. The father protecting his family from the cobras? Good. The family invading the cobras' land by force? Eh, not so sure about that.
And in either reading, the shotgun and stick represent man's power over nature—the power of death.