Couple things here, why doesn't 'he lives in the house' work here?
Also, I've got confirmation from a native speaker that 'He stays at home' is an awkward translation compared to 'He lives at home', which seems to be the more common usage for this kind of sentence and should at least be an acceptable variation.
I don't think the translation is awkward. "Tinggal" has two meaning/translation: "stay" and "live"
It means "to stay" if, for example: my mom, my dad and my sister left home to have some dinner. But I "stay at home" (tinggal di rumah) because I am sick. This phrase is commonly used for this context.
"Tinggal" has one more meaning that is commonly used. But I don't think we have the word in English so I'll just explain:
I was running late to work, so I was in a hurry and I forgot to bring my wallet. So we usually say:
Dompet saya tinggal di rumah.
So, it's like saying: I'm going out, but my wallet accidentally staying at home
Does it mean "he stays home" as opposed to "everyone else goes to the party"? That's what I thought the Indonesian meant. In that case, in Australia we would more commonly say "he stays home" rather than "he stays at home". But if the sentence is about where he lives, then in Australian English we would never say "stays". Always "lives".