Have you found out what the cost of sending your child to LA school is compared to the cost of the independent, factoring in different transport costs (presumably lower at nearby school) Is that an argument you can use? And terms are shorter at indie, meaning again lower transport cost.
Also stress to child of further school, long day etc?
These all could be used as arguments at Tribunal, but I do not know what your Indie costs.
If you look at your child's assessments/professional reports from Ed Pysch OT etc, are there any sentences that you can use to prove that the LA school is unsuitable, whereas Indie school is more suitable? Peer group, specialist teaching, small classes, environment, curriculum? I remember someone at IPSEA saying to me she had won at Tribunal because the indie school she wanted had a farm, and she was able to use the evidence that her daughter needed a holistic learning environment to argue that they needed a school with a farm! I think you have to go through with a toothcomb looking at the provision in both schools and say why one is not just more suitable but why only one meets the needs of your child. Something about the LA school will not meet her needs, that is what you have to pinpoint, whether it is anxiety over long journey, outside local area for friendships.
This is what Parent Partnership advised us too, focus on what is only provided by the school you want, and make sure her needs are defined fully that the school you want is the only school that meets those needs.
For example when we didn't want ds to go to x school we pointed out that there was no clear behaviour policy on the school website that ds could understand (it was all very woolly and talked about responsibilities and community), and that he needed very clear rules whereas the school we wanted had the behaviour policy in black and white (ie when you chew you get a negative point, 3 negative equals a detention). They were both mainstreams but that is a good example of how we worded our request for y school, even if our real reasons were far more of a "gut feeling" - x was raucous hurly burly and y was a calm ordered atmosphere. We didn't say that. We talked about no of kids in x and no of kids in y (far less)
good luck.