If you end up at Tribunal to fight for an indie ss, then you can "win" it on one of two grounds.
a) That the school the LA have named cannot meet your child's educational needs
OR
b) Parental choice is cheaper than the LA choice. When doing the costs, take into account the cost of the LA place (which is normally just your LA's AWPU - there is established case law on this, despite all the recent funding changes). Also addin any "bought-in" provision (eg OT/SALT/TA/LSA) & any transport. But the transport is only if the taxi/minibus/etc to LA school isn't already going there. So, for example, if there is already a taxi going from your area to the LA school, then it'll either be a zero cost or the precise amount the taxi firm add onto the bill specifically for your DC. Or, if there is no child going from your area to the school, then it'll be the whole cost of the taxi.
Likewise with TA/LSA support - if Statement has 1:1 TA then you can add in 100% of the TA cost, but if it's not specified that it's a 1:1, then it'll assumed to be the regular class TA, so you can't add any TA cost in.
However, if you can establish a) then you do not need to worry about b) at all.
So you must establish that a) is the case - the LA named school cannot meet needs. You will also need professionals to help you with this. You may need an indie EP to go into the LA school and write a report about the LA named school compared to your DC's needs. It's also helpful if any LA report confirms something like small class size/peer group of similar children. Whilst a LA report will never state indie school, they might state small class size or peer group of similar pupils - my LA's EP report stated both but the LA ignored it - a fact the Tribunal's Judge picked up on and commented on against the LA. So look out for any phrases or words in the LA's reports that could support your case.
Also, visit the named LA school with a very open mind and ask very pointed questions about your DC's needs compared to what they can provide. Afterwards, document the visit, and if possible, get them to sign it that it's a true account of your visit (the school will probably not do this). If you get to Tribunal, ensure that your visit report is included in all your extra evidence. I did all of this and my visit report became a crucial piece of evidence during the Hearing because it showed that the LA school had no experience whatsoever of my DS's needs and that the teacher had admitted this to me before she realised there was going to be a Tribunal. The school hadn't signed it, but during the hearing, the teacher (who was a witness for the LA) agreed in front of the Judge that she had personally answered all the questions on my visit report. It proved more than even my indie EP's report that the school couldn't met need because the teacher had answered it honestly without realising the implications of a Tribunal.
When working on the Statement, there is established case law that you do part 2, and only when that's right, do you do part 3. Finally when part 2 & 3 is right, only then can you do part 4. However most LAs work from part 4 and fit everything else around it - this is wrong with clear case law showing this. Remember that part 2 is of equal importance as to part 3 - in fact it is probably of more importance. Part 2 is like a "doctor's medical diagnosis" so should state needs (or diagnoses), whilst part 3 is like the "prescription" to all those diagnosis/needs. So you can't have a provision in part 3 unless there is a clear "need" (or diagnosis) in part 2. If you DC has many needs/diagnosis then the Statement should show how complex your DC is. If your child ends up with much provision quantified and specified in part 3, then this too will help the case for an indie ss. My DC's Statement has over 17 hours a week of q&s external support (SALT/OT/Specialist dyslexia support) - a fact picked up by the Tribunal that this could not be done in a mainstream school without serious impact on my DS's access to the national curriculum.
Whilst it is important that the indie ss can meet your DC's need - ironically this is of less importance as everything stated above. For my Tribunal, all we had was a letter from the indie special school saying they could meet need and how/why they could, along with Ofsted reports & school brochure. No EP (either indie or LA) went into the school to determine if it could meet need - although I found out afterwards the LA should have sent someone in - in which case I would have to have counter-acted it with my own EP going in.
It really is important that you try as much as possible to prove point a) above. You will need indie experts and it's possible you'll end up at Tribunal as must LAs routinely fight indie school placements. But it can be done. I did it, as did most of the DC at my son's school.
Good luck! It's a long, hard fight, but it can be done.