Would it strengthen my application if I studied for an LLM in addition to my LLB?

Studying for an LLM would undoubtedly increase your level of legal understanding and knowledge, particularly if it focused on a particular area of law competition for instance. This might be an area that you have a strong interest in and want to learn more about and BLP would certainly look upon this favourably. It wouldnt, however, mean your application was necessarily stronger than a candidate who had just an LLB, as there are many other factors as well as academics that would be taken into consideration. Work experience, for instance, is just as important as further legal education. You need to consider what you want to achieve by completing an LLM and what your aims are long term, ie particular areas of specialisation, career path. You shouldnt simply study for it to increase your chances of getting a training contract, it should be something of real interest to you.