Hey hey now, don't forget that MC is the only one in the story with "future knowledge."
Blasting Han Byul for making the safe choice, after her entire world got turned upside-down, seems kinda short-sighted.
We've seen MC face down a horde of zombies and an Alien on steroids,
We know that he's a badass from the future that came back in time,
but no one else knows what we do.
Please, please, please, don't mix the reader's knowledge with the character's knowledge.
In chapter 10, she begged him to trust her, and he didn't reciprocate in any way.
"Han Byul, I . . . " (unless of course they had a secret conversation off screen in which he explains everything, and she's gonna be his spy in Golden Lion, but is just pretending to be unhappy with him to convince anyone watching)
Then in this chapter, he expects her to trust him, no question?
"I thought I could at least become a sturdy wall you could trust and rely on, but I guess I'm not."
"I've heard you've barely been talking to the others these past few weeks."
Sure, Han Byul hasn't been talking to the team much, but that goes both ways.
"I've heard . . ."
MC hasn't been seeking her out / communicating with her either.
"Who are you to treat them like they're just deadweight?"
Pfft, MC did it first. Run away from the zombies, i'll take care of it. Run away from the scary monster, i can beat it but i'll let you feel guilty about it instead.
On one hand, we have a well-established guild willing to bring you into a position of power after recognizing your talents.
On the other, we have a shady dude always telling you to run away, because you staying will only be deadweight for him.
Honestly, I'd probably stay with the Lions too.