Thanks for the ask! Hope you enjoy the answer + any justifications. For the 9 LIs + Kyo and Amagiri (I haven't seen enough of Iba, Souma, and Sakamoto):

Saito: “There are still good people in Slytherin...” The sorting hat muses before asking him, “Where do you think you belong more, Slytherin or Ravenclaw?”

“Wherever I am ordered to go.” Saito responds demurely.

The hat gives something of a sign. “Of course. Ravenclaw.”

Souji: Souji can’t stop grinning while he is being sorted. He knows wherever he goes, the fun goes. He now has access to magic and is gonna make Hijikata’s life miserable. The world is at his fingertips. “Oh?” The hat senses his boundless need for mischief. But also senses the same lack of goals just for his sake (and not Kondo’s) and lack of sense of self like Saito. And thus, he places him in Hufflepuff.

Chizuru: Hufflepuff. Not that Souji is complaining. He grins even more. Now, what shall he do~?

Heisuke: It would be a toss for him between Hufflepuff and Gryffindor. He whispers very quietly to be placed in Hufflepuff, and the Hat knows it’s because that’s where Chizuru is. “N-not that it matters much!” Heisuke says. “...But it would be kinda nice, you know?”

Kazama: Without a doubt, Slytherin. Kyo would also follow him, but Amagiri would end up in Ravenclaw, and have several classes with Saito, whom he respects as a peer.

Hijikata: Slytherin as well. In game, he’s not afraid to be portrayed as a bad guy in order to get the job done, and he’s willing to have a notorious reputation if it has the most overall benefit. He also is capable of fighting dirty according to the baka trio. However, his placement soon causes a lot of commotion between him and Kazama. The detest is mutual. Hilarity often ensues, but sometimes their duels need to be stopped...

Sannan: Ravenclaw, but the hat notices his thoughts need to be kept in check, because otherwise he has the potential to become on par with the most ruthless Slytherins. The hat tells Sannan as such before yelling his decision. To which Sannan gives a small smile. “We shall see, now won’t we?”

Harada: Gryffindor, but is popular among the ladies in all of the houses, especially Hufflepuff. Chizuru, however, is fairly oblivious to his reputation.

Shinpachi: Gryffindor

Yamazaki:  Gryffindor. He is obedient and doesn’t think for himself to a fault, just like Saito. However the hat acknowledges his immense potential to change himself when he is not busy dying in everyone’s routes, which is unlike Saito (not to his extent, anyway).

However, the Shinsengumi would not be the Shinsengumi without each other. Despite their houses, they spend a great deal of their free time together, becoming well known in their time at Hogwarts for being the largest inter-house friend group. Well, except for Kazama. They think he’s shit.