First, if the company in the original post is a state-owned or unknown company, it will be a very hard case to sell regardless of your title. It’s a KNOWN fact that top business schools prefer people from famous foreign consulting companies or multinational conglomerates. In other words, it’s very important to have experience working in a recognized and respected international working environment. Coincidently, most of the candidates with this kind of background have better English communication skills as well.
Second, no, LBS isn’t much better than Oxford or Cambridge. Take O for example; when you compare the vital things, such as employment statistics, student quality, faculty quality, and brand name, Oxford is not behind in any of the above. Sure, LBS is currently ranked higher and is more recognized among employers mainly due to its longer history. | 
