Although it was written with 11+ in mind, I assume you've read the following?
http://www.elevenplusexams.co.uk/11plus ... rs.php#b11
For 13+ one would hope to see level 5s in English, Maths & Science at KS2, and I suggest he should appear to be on target for at least high level 6s, preferably level 7s, at the end of KS3.
How important is it to provide extenuating circumstances if the score is so close?
I don't believe it is.
Any idea what the success rate is for these kind of 13+ appeals?
What you really want is the success rate for a 13+ appeal with 120. We've seen such a statistic for the 11+ before, but I'm not aware such detail has ever been published for the 13+. I suspect you'd stand a good chance, provided you have strong academic evidence.