We are agonizing over JAGS vs Alleyn's, but I wouldn't agonize over JAGS vs CLSG (although it's very easy to say when you are not in those shoes!) if commute to JAGS was easier and you have an academic scholarship to JAGS (an honour and looks good on CV).
I simply do not see how CLSG is in any way better to justify the commute. Academically, CLSG and JAGS are identical (CLSG is #3 in Times league tables and JAGS #9), with some subjects JAGS outperforming City on GCSE a lot (e.g., History and Geography), similar in maths (82% A in City, 80% at JAGS), similar numbers to Oxbridge (12-15 depending on year), etc.
If anything, I'd probably say that JAGS is better because they achieve this with a slightly less selective intake - 800 applied to City this year vs ~480 to JAGS.
In terms of social mix/values they are similar - diverse, multicultural, etc. Don't know about City, but at JAGS 20% of kids are on bursaries.
JAGS has better grounds, fantastic language provision (Italian, Japanese, Russian taught) and a headmistress who is a know quantity whereas at City there is change which may well be fantastic but a bit of a risk (as with any change).
Plus JAGS has a bigger year group which at secondary is an advantage, in my view.
We, on the other hand, are agonizing over JAGS vs Alleyn's (academic scholarships at both) and, I think, it's a more difficult choice since JAGS and Alleyn's have a bigger gap academically AND you don't get as much diversity (both social spectrum and multicultural dimension) at Alleyn's, but we like the co-ed concept. Commute time, obviously, is not a factor since they are opposite each other.