As a very recent winner of this achievement, one thing that would've been very useful to me and doesn't seem to be on any of the guides or videos, are the times needed to trigger the checkpoints in no fighting in the war room. Particularly the first three where the combat is most challenging.
The times for each are given below along with the time I took to show the amount of leeway you have to get there. Also I had the SAW from the previous level which I found really helpful along with the grenade launcher attachment on other weapon
Checkpoint 1 (top of the stairs) 7:30. (My time 8.10) Though this one is tough it's not as tough as later ones. Use this checkpoint to get a good starting time 8.30 would definitely be possible.
Checkpoint 2 (before the dreaded 3-way corridor of death) 6:30 (my time 7.12) There's no two ways about it, this is an absolute bitch. Giving yourself enough time from the previous checkpoints will help so you can take this section slower. You need to activate the next checkpoint by 4:30 so take your time in this part and use flash/ nades and launcher alot. Use each death as a learning experience as enemy placements are fairly consistent
Checkpoint 3 (entering the blast doors as it shuts behind you) 4:30. (my time 4:35) Once you've got here and hit the checkpoint you've basically completed the level
Checkpoint 4 (blast door about the open) 4:00. Final couple of rooms to clear but compared to what you've just been through, the rest is a breeze! There are a few more flash in the room before you enter to help.
Checkpoint 5 (ready place place c4) 2:50. The last group of enemies are easy pickings with the support you (finally) have
Checkpoint 6 (enter the codes) 1:00. Well done! Time to get outta there and go mile high