(Couldn't fit this into a comment….)
I think that, in this story, the cat primarily serves the function of atmosphere (that of having something else in the room rather than just Egbert and Lady Anne); otherwise the (already very short) story would be even shorter, and boring. I don't think any deeper symbolism needs to be looked for: while, as far as "philosophy" goes, there is no doubt a certain spirit and attitude towards things that pervades Saki's stories, it appears he wrote primarily to entertain. But also:
This picture of family life (the couple had a bird and a cat, and presumably no children) says something about them. (Some reviewers have expressed the opinion that Saki was disdainful of childless women.)
Many of Saki's stories (e.g. Tobermory) have the idea of animals (and cats in particular) being superior and more intelligent than, if not most people, definitely the kinds of people Saki was contemptuous of (as here he is of their artistic taste and intelligence). So here, making explicit the superiority of the cat (the line about "superb indifference", the cat refusing to act as a vacuum cleaner, and finally the fact that the cat, which thinks of Egbert as “A fool”, seems to know of the death already, unlike him, and is only waiting for his exit) expresses the character of the couple better.
The punchline becomes stronger: The penultimate sentence, “[The bird] had cost twenty-seven shillings without the cage, but Lady Anne made no sign of interfering” sets up the idea that while a lady like that might well ignore her husband after a quarrel, to ignore the loss of an expensive purchase— now that is truly unusual and demands an explanation, and this expectation is answered by the final sentence: “She had been dead for two hours.”