Shot at LSE Law?

Law at the LSE is supposedly heavily oversubscribed. I'm wondering whether I have a decent shot at being made an offer - I'm in a position where I have 5 other respected universities and am content with keeping them, if not for the fact I may change on to the LSE, if I have a shot at gaining entry.

The real problem is the following: I have mitigating circumstances for my GCSES in which I achieved 10 A - C with one D grade. Pretty bad right? I can't do much about it now, but that's that. Many more if not at least 90% of applicants applying for the LLB course will most likely have better GCSE grades than me, so competitive-wise GCSEs are off the table.

Apart from the above, I have achieved:

Achieved AAAA in facilitating subjects at AS level.
Predicted A*A*A*A*

Good personal statement in both mine and my teacher's opinion.
( I understand this is vague.)

Good references

Another problem is the fact the LSE does not require the LNAT - so differentiating stronger candidates is made much more difficult.

Help is appreciated given my time limit. Especially as I understand how annoying these threads are.