Kakashi Hatake: Finding the Way on the Path of Life

Kakashi wasn't...couldn't lose Rin after losing Obito and promising him that he would protect her.

So when the mist-nin had spilled that the Sanbi was sealed within her. He decided not to tell anyone, including Rin herself. He couldn't fail Obito again. Not like this. Even at the risk of the entire village, he wasn't going to break his promise.

This changes everything.

Or:

Rin doesn't know that Kakashi knows she's now the carrier of the Sanbi, and with the enemy-nin chasing them, she's more inclined to bring Kakashi back to Konoha safe, than anything else. As for killing herself? She'd rather serve as a meat shield for her friend, and she could always go with a kunai to the throat as soon as Kakashi turns his back.