All of the pronoun contractions are somewhat inconsistent with noun possessives. So "its", but "the dog's". There's no reason why you couldn't write "it's" for both "it is" and "its". You already write both "the dog's ball" and "the dog's tired", and both "it's" and "it's" are pronounced identically, which doesn't pose any difficulties in speech.

His, her, and my are a different matter, since they are all pronounced differently from he's*, she's, and I'm.

* at least in most accents, "his" uses the kit vowel, while "he's" uses the fleece vowel