Alignment is a tool, not a straightjacket
I will lean in support of those who say Lawful Neutral. You can say "Lawful" doesn't apply to personal codes, but I think most folks will disagree with you. "Chaos" is "do whatever I want in the moment" so a personal code definitely infringes on that. In the end, the alignment chart is what the DM says it is, so your mileage may vary. The DM of a group where I'm a player doesn't see alignment the way I do as a DM, but that's okay, it's his game.
I like the fact that the alignment chart gives players a helpful way to figure out how to play a character, but I don't like when it feels like a straightjacket limiting a player's well-thought out personality for his character. This PC sounds like a D&D version of Jack Bauer from "24". In fact, the part about torture makes me think that's probably where he got it from. If that's the case, I'm more certain that it's LN, because that's how I see that character. And it makes what everyone else sees as "moody" and puts it into perspective. He's out to save as many innocent lives as possible, and that calculus is all he worries about. (Math is very LN, and it's usually not a man-made legal code.)
That all said, here's some constructive advice: there's no need to struggle with this, since the alignment chart is usually just a tool to help role-play, as I've said. In many cases, a character changing alignment only affects how you interpret certain rules, like if a paladin stays in the good graces of his religion or in the employ of his order. And if he doesn't change alignment as you've decided it, it doesn't really matter. Figure out how much that's really in play in your game, and for this character, and feel free to be flexible. Maybe make a house rule that certain characters are exempt from "alignment" per se, as long as they play the character they describe. But if you really want to define a rule to help make sense of things, flip things around and make the alignment chart bend to the actions, rather than the characters bend to the alignment chart. Specifically, use an Alignment Tracking system. I use that in my local in-person game and it's been great tool to help my players (who are new to RP) better navigate what alignment is all about.