Funabori would pull her own weight, she's excellent at housekeeping.
She would stay in the girls room, not invading Kazama's territory like Ataru.
Her reasoning is sound, its late at night and its safer for Funabori to stay the night.
And that is Kazama's house, so he doesn't even have to ask whether or not the other girls are okay with it.
So yeah, i see no reason for Kazama to say no lol (even though it does smell a bit like PLOT)