"That's what my character would do" isn't actually the root of the problem, but a symptom of it. I mean, isn't everyone just doing what their character would do?
The root of the problem is creating a character that doesn't have the personality to be a functional member of a group, which is often a symptom of being the sort of player that doesn't have the personality to be a functional member of a group.
The correct response to, "That is what my character would do?", is "Why did you create a character that was going to not get along well with others, and in particular wasn't going to get along well with this party?"
And stepping back from that further, if it was obvious from the character sheet, character concept, and backstory that this PC wasn't going to play well with others, why did the GM approve the character? This is absolutely especially true when dealing with a player you haven't seen play before.
One GMing mistake that took me nearly two decades to really overcome was the idea that a player could play anything that they wanted, and I as GM didn't have anything to say about that. This mistake was rooted in an idea that I still consider a very good one, which is "Everything in the game universe is the province of the DM, except for the PC's, which are the exclusive domain of the players. GMs should never tell player's how to play their characters." But, the problem with extending that out to the level of character creation, is that it's also the responsibility of the GM to make sure the game is a cooperative, social, enjoyable play experience for everyone, which means that the general idea of the party or group has to be established. Is the party generally heroic? A villainous character might not be the best idea. Is the party generally villainous? Well, an honorable character might not be the best idea. If a player is going to play a character that will prompt them at some time during play to say, "Well, it's just what my character would do!", chances are you should have put your foot down when the character concept was introduced.
And if, "That's just what my character would do." persists even then, then probably the best thing to do is make sure the character bears the full weight of their choices, and tell the play privately that the group is not happy with the player choosing to sabotage the story and they need to find a motivation or character for continuing the story rather than doing stuff that puts a stop to it.
As for this particular situation, more than 6 players can be tough to deal with. Six is I consider the top limit where the actual personalities of the PCs can matter. Above six players RPGs increasingly have to focus entirely on group goals and personifying and exploring character has to recede into the background as no longer a viable aesthetic of play. There just isn't enough time to devote to individual spot light or individual goals. If your dominate aesthetic of play tends to be puzzle solving, tactical combat, and stuff like that, and everyone is generally OK with a story on rails with a single group goal like, "Defeat the BBEG and save the world", then you can generally do 7 or 8 players. But there won't be a lot of deep and meaningful RP, because it's really hard to have 8 or 9 way conversations. In my experience, you are pushing to the point where groups tend to fracture, pushing you toward a situation where you are trying to run two or more simultaneous games.
The other problem you get above 6 players, and really this is already a problem at 6, is that if you are running any kind of story based game, you really can't afford to have players missing from a session. So the more players you have, the harder it is to get everyone together. Once you get up to like 8 or more players, I find that about the only game you can run is a haven/delve type game where at the end of each session everyone is assumed to retire to the haven. That way, anyone that is missing can be assumed to have returned to the haven and had some business to attend to. It's almost impossible to run a game successfully in the long term were the PC is in the session, but the player is not. "Someone run my character for me" does not work 90% of the time.