First of all you're underestimating that Quincy(I forget her name) by calling her a "little girl", after all she's got captain level power.
If he had used Hinamori and she took that attack and didn't die everyone would be crying foul about "oh my God how can she take and survive an attack that Komamura, a captain, had to change his plan of action just to defend correctly so he wouldn't be severely damaged". I guess in the end it's no-win situation, people hate that Shinji got one shotted, and they'd also hate that Hinamori took such a powerful attack without dying(I know I wouldn't like it). Plus Hinamori will probably heal Shinji, that's another reason not to use her as the guinea pig, as I feel everyone else that can heal probably has a role to play elsewhere in the war. At the end of the day Kubo knows what's coming up as it's his story, so we don't really have the right to question every little decision he's made until we've seen enough to suggest that he made a bad choice(which of course he has in the past, but that doesn't mean he has now despite certain opinions). I mean of course it's Kubo's fault for giving Shinji that kind of stupid arrogance that he gets whenever he uses his Shikai which lead to him getting dropped, but what can you do? I'd like to hope next time he uses it it ends up being useful lol.