In short I would say wait a year and don't make any applications this year. If you get a 2.1 or a First in your finals this is all you need to put on your CV (firms very rarely ask about your results in earlier years, and at the end of the day your most recent performance will be the main determinant of how they view you).
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If you apply to the main firms right now, you are likely to get only a couple of interview offers at best - there will simply be too many Oxbridge and LSE candidates with I / II.I results that will catch the eye at your expense. If firms have rejected you once it will make it that much harder to be accepted in the future, so you are better off not starting to apply until you can put your best foot forward.
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Having said this, do take the opportunity to attend careers fairs, company presentations, etc. while you are in your final year at LSE. There will never be a better time for you to get to meet the firms and decide which you like best. But spend the time you would have spent on applications and interviewing... on making sure you get a First or 2.1 in your finals!
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As to what to do in the intervening year, you want to have something worthwhile on your CV for this time period. MAs typically enter consulting firms at the same level as BAs, so I would not take this option unless you have a particular interest in the subject you will be studying (and so are doing it for personal interest rather than as a career move).
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The single biggest thing you could do to enhance your CV to appeal to consulting firms is to get yourself a(nother) foreign language you are pretty fluent in. A year spent say in France or Germany really improving your language skills will get you a lot further than an MA in a similar subject to that which you've already studied.
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Hope this is useful advice and good luck with your career move
