Well we are assuming the space cat (i.e. it is not an ordinary cat) emitting a white light (it’s a glowing cat!), that is, the peak is in the middle of the visible spectrum. The cat becomes closer to red as it moves away and to blue when it is moving towards us. If it goes fast enough, the peak shifts to IR (past the red) or UV (past blue/violet)

But yes, if it was a normal non-space cat, the peak would be in the IR normally (since cats are quite cool, in both senses of the word), and as it moves away the peak shifts to longer wavelengths, and as it moves towards us, it will go to shorter wavelengths, possibly becoming visible if fast enough.