OP, do also bear in mind that 11 plus boys places at LU are probably more competitive than girls places because most girls applying to LU will also be applying for G&L and SPGS as well which are just down the road. For boys, there’s only Hampton which is miles away and the whole coach thing doesn’t appeal to everyone. Or KCS which also seems quite far out to more central London families. SPS still has very limited places at 11 plus.
Either you meet the bursary criteria or you don’t. If you do, apply.
There is no stigma at all to being on a bursary place. In my DS’ friendship group, there are ones who live in Mayfair and others who live In towerblocks adjacent to Grenfell It’s extremely diverse, but it’s an urban school which reflects London because this is London. The school won a Times award for enabling social mobility, alongside the NHS and the Home Office. I have also had / have children at G&L and SPGS and these schools are less socially diverse, tbh, but trying to make steps in the direction of LU to be more accessible - eg. G&L are now running Saturday classes for local primary school children. This is something LU have been going for some time. At G&L, 10% of children receive bursaries and they are looking to increase this to 25% which is where LU are at. G&L don’t offer academic scholarships at all; there’s only one music scholarship at 11 plus. Following the Grenfell fire, LU accommodated the sixth formers and teaching staff from a college damaged by the fire, so they could finish their A-levels. DS has friends who have lived through war zones in Iraq who researched and wrote their own applications for themselves and brought their younger brothers / sisters to the exams to give them a shot. He has friends who do the entire family weekly food shop on their way home and have to take on far more responsibility than the average 16 year old, yet when they tell you they want to be a brain surgeon, you don’t doubt it. They all learn a lot from each other, they really do and it helps them to appreciate the opportunities they have, whether their parents are paying or not. The best thing about the school is it’s diversity and I think you have to embrace that and be honest about your own situation when you apply. If you won’t consider school fees, then either apply for a bursary or don’t apply, but don’t put him up against thousands of others on the off chance he gets this headteachers award (whatever that is) topped up with a music scholarship. Lots of children apply for music scholarships, but get told not to bother nearer to the auditions, even though their DC might be grade 6/7 on a few instruments which is very good by any standards at 11.