For something like this I would use a system like Torchbearer, a game based on both The Burning Wheel and Mouse Guard - Role Playing Game (Cannot post the link, sorry).
Torchbearer has been designed to be a modern take on the original D&D, making it very lethal. I have read the rules and I know pretty well both Burning Wheel and Mouse Guard. They have a very interesting combat system and Torchbearer makes use of Mouse Guard's simplified combat system making it a very fast strategical combat where precise positioning does not matte but trying to predict enemies' behaviour is crucial to survival.
Magic is not a crucial aspect, while careful planning is. And the game does not expect you to narrate a lot. It is very focused on exploration. As an example: going to a town, for the game, is just a play phase, in which you decide the actions, roll dice to fix things and replenish your equipment and then the group is back in action. It is stated that it should not take more than half an hour.
I also consider GURPS 4th a medium-complexity game and very close to what you are looking for, but I know that most people consider GURPS very complex, so I would say you should take a look at it but do not loose too much time on it ;)
Since I have been asked here you go some of the experience I have with the listed systems:
The Burning Wheel
I have ran a one year and a half campaign using The Burning Wheel plus a couple one-shots, either with high magic, almost no magic or steampunk. The system allows players to have magical backgrounds, but using magic is extremely dangerous, so most players and DM stick to no-magic settings or low-magic settings. However no character is required to know or have something to do with magic. In addition some of the backgrounds are related with a negative view of magic, so I suppose this would be good for your case. Another thing that adds something good for your setting (if I remember correctly, as I read only the first novels) is that characters require A LOT OF TIME to recover. A LOT. This can help you in advance your power struggles while characters wait to recover.
The Problem with The Burining Wheel: It's VERY complex to learn the fighting system. If even one of your players do not know the rules by heart fights become long. And there is a problem with fights different than one-on-one (i.e. as you can split multi-character fights in separated one-on-one fights, you might end with two-on-one, three-on-one, etc. fights. The "one" in the last fights WILL DIE. No exception. I think it's realistic, but players do not want that, I suppose) or combining melee and distance weapons.
I ran 2 short (2~3 sessions) campaigns using Mouse Guard. Mouse Guard takes The Burning Wheel background and combat system and removes all the complex bits, leaving only the interesting parts. It is very streamlined but also technical and can be lethal if players are not careful (in my first campaign one expert player died at the first encounter, and I set it to be a tutorial), exactly what you were asking (Quick combat, low/medium ruleset, lethal if not careful). The game also integrates a VERY interesting season mechanic, but I suppose we do not want Winter to come, do we? ;)
The Problem with Mouse Guard: Players should be mice.
I managed various campaigns (I started playing 14 years ago with GURPS 3rd and participated in more than 10 GURPS campaigns, one of which was set in Westeros even BEFORE all the social buzz. I think it was 2006) and planned and ran a convention one-shot with GURPS 4th. As I told you before GURPS 4th can be pretty complex if you use EVERY single feature, but you can skim it by removing some of the things that make combat complex (such as the over-complicated distance modifiers, which require you to keep track of every single yard of distance between fighting characters, or some of the most complex fighting actions). GURPS 4th, however, is pretty brutal in fights, making every encounter potentially lethal, has NO classes (only points to distribute to create characters), high customisation potential and magic is viewed as just an add-on to the game, so you can remove it without changing anything/
The Problem with GURPS 4th: PowerPlayers could come up with some over-powerful Advantage/Disadvantage/Skills combination. Keep them in check. e.g. Do not allow them to get a talent in weapons, it will destroy your campaign setting, I think it will result in some anticlimatic boss battles.
I still have to try Torchbearer due to the lack of time and because I have been away from my group since I have bought the manual, but I have read the manual and having extensive experience with the parent systems I can suggest them for this scenario, as Torchbearer gets the streamlined fighting system from Mouse Guard and integrates it in a gritty fantasy setting.
About the setting
As you might have read I have played a short campaign set in Westeros and I have just ONE SINGLE suggestion to you: do not play campaigns based on very structured settings. I do not know which players will play with you, but if you have a difference in setting knowledge this will ruin the enjoyability of your campaign. People will try to connect to the novels, others will just play because it looks fun while others and others will have problems following all the details of the setting. For a short campaign come with your take of that world, detach it from the novels and the TV series, you'll have much more fun this way.
This is a personal point of view on how campaign settings should be chosen and diverges from the suggestions I have provided you. They still persist if you'd like to run a Westeros campaign.
An additional system you might want to consider - QIN
Since you want to play a Westeros campaign I did not suggest an additional syestem which looks it reflects most of the points you listed, but I will write a couple of things here if you are interested:
QIN is an RPG designed to play Wuxia campaigns, however the "wuxia" powers are very limited in the first levels for character and the setting is very low-fantasy and gritty, with some fantasy creatures creeping around, but most of the enemies are humans. However the game is set in ancient China (or in ancient Eastern setting), so it might be somewhat complex to manage a QIN campaign. I have played a couple of short campaigns and I have to say I really had fun with it. Take a look at it if you'd like, you might find some good ideas to use in other gritty Western fantasy campaigns.