The English reflexive isn't actually a reflexive. It's an intensive, as evidenced by the fact that it's different from the oblique in the first and second persons (me vs. myself, you vs. yourself) unlike many other languages (mich vs. mich in German, for example). It's used for the reflexive because English lost its original reflexive, and the first and second persons were dragged along by analogy.
As a result, there are many, many cases where one would expect to use a reflexive in another language, but use a simple oblique in English, even in the third person. This is one of those: since we don't particularly care about the house (it's the over-all topic, the cat, as new information, is what's really important), we use the simple oblique "it" rather than the intensive, and incidentally reflexive, "itself" to avoid bringing attention to it.
This is especially true given where the "itself" would be. There are two places an intensive pronoun can be placed - directly after it's antecedent, as in "I myself will do it", or at the end of the utterance, as in "I will do it myself". Because your proposal would put the "itself" at the end, it would be an intensive regardless, and you might as well say "the house itself has a cat in it" - note that we still don't use the reflexive in the last position!
Aside: The reflexive possessives are more transparently intensives first, as they are open compounds of the possessive pronoun and the word "own", rather than closed compounds of pronoun plus "self", and we rarely use them unless we are being intensive, rather than reflexive. I bet you didn't even know English had a reflexive possessive until now