Paraphrased:
"You can bring anyone."
...
"Not these ones though."
I get strong vibes of "I am altering the deal, pray I don't alter it any further."
That's cheating quite a bit, but I can accept her reason to exclude the two demons.
Excluding Shiva after explicitly citing him as an example of person he can bring is not right though.
Asking for his identity is a bit much, but she could have asked him to prove that he's human at least. After all, Haruto brought two demons as team members, so who's to say that Shiva isn't also one?
So, who's left? Assuming only characters we already know: daddy, mommy (not sure about her fighting potential, but she did volunteer as an attendant earlier), sister, teachers (the tiny one and her rival), royal siblings, the totally-human-not-former-demon-king friend... and the principal herself. If we're stretching things a bit, maybe he could even get the queen to "volunteer" in exchange for removing her collar, but I don't think she'd be a reliable ally. Also the PE teacher, but I don't think he's sufficiently connected to Haruto.
Most likely, Iris will make the final list. And tiny teach could suggest to include his sister as a form of credentials so she can join the Academy early. If we stick to a team of four, the last spot could go to the prince or the principal.