I am a military professional, who has worked in both Joint and Multinational environments at a pretty senior level (3 & 4 star - albeit as a PowerPoint monkey), with both government and private sector defense "thinkers", so I hope that I can give you a customers eye view of your (very enviable) choice.
Bottom Line Up Front - My view is that you would have to be insane to choose anywhere other than Oxford.
The LSE is, rightly or wrongly, going to have to work tremendously hard to overcome the massive damage done to its reputation in the defense sector (particularly in the US) by its links to the previous government of Libya. It has an excellent (although tarnished by its apparently flexible view of what constitutes academic standards) academic reputation in the field of IR. In the field of defense it has had little of any value to say on the major themes of the last decade; that is not to say that those working there haven't published good stuff, but rather that what they have published has not been influential.
KCL is excellent but has massively undermined the value of its own unique selling point by taking on too many average army officers to do postgraduate work. It has become, in the view of many in the US defense establishment, the equivalent of the Security Studies program at Kansas State, which leeches off the US Army Command and General Staff College. Within the UK it is not regarded as highly as Oxford, even within "those who understand these things" because those who understand these things know that Oxford has, under Hew Strachan, the Changing Character of War program, which is frankly well above anything offered by KCL. Unless your PhD is on a particularly esoteric topic, like the military thought of Fuller, then everything is better catered for in Oxford, which also has more senior military folk knocking about the place.
Finally one last point. if you walk into a room in Kabul and your resume states that you went to Oxford, you will be listened to more than if it says KCL/LSE. That is not fair, nor is it always helpful. But it is true.