I would recommend considering doing the General Course for your third year... it is a year study abroad program. Often, if you talk to your advisor at LSE once you get here, they can make you a conditional offer based on how you do during your General Course year.
I am currently trying to do this, as is someone else on this forum. I would also like to transfer in to LSE as it is a much better university. I haven't herd of many students from other schools getting straight offers to transfer. But, I have herd of students transfering after a GC year.
If you have any questions, please PM me, I can always keep you posted on how things progress once I get there and talk to an advisor.
Good Luck if you do try!