Unfortunately, slow progress, on its own, won't swing it at all - although it can add weight to the rest of the evidence.
You need hard written evidence from professionals as to exactly what needs your DC has, along with them quantifying how many hours therapy/teaching etc DC requires for each need.
Then, after you have that, the FOI is your friend to determine from the LA school if they can meet those needs
For example, one of my DS's diagnosis is severe dyslexia - agreed by 2 indie EPs and 2 LA EPs so that was never in dispute. Although 3 EPs said dyslexia was primary barrier to learning but 4th (LA EP present at Tribunal) said difficult to tell whether primary need was dyslexia or speech and language - but then promptly stated primary need was speech and language (judge tore into her for doing that and EP totally undermined her own evidence because of it!)
So next stop was FOI sent to the school asking questions like
- how many teachers have a qualification for dyslexia
- how many pupils have dyslexia on their statement
- how do they differentiate for dyslexic pupils
- and the same questions again for each of my DS's needs
Etc etc
Surprise surprise FOI came back with no dyslexia experience, no teachers with dyslexia qualifications, no other pupils with dyslexia. So when the LA lied were economical with the truth during the hearing, we were able to go straight to FOI (which was all in the evidence bundle) and look at the FOI to show they were lying mistaken
Also, the FOI can also point out other lack of resources. For example, the LA school had an attached SALT unit - the FOI showed it was under resourced and they would have to buy in help for DS. Even worse, the FOI showed that despite it being a SALT unit, no SALT had stepped foot in the place for a month prior to the FOI request. So it was very under-resourced and for anything quantified on DS's statement, it would all have to be bought in to guarantee he'd get the provision.
I also visited the LA school with a totally open mind and formed my own opinion and wrote a very business-like report. This report included the fact that the teacher in charge stated to me that they had not received all the reports to DS's Statement and were therefore totally unaware as to the severity of DS's dyslexia and related anxiety. So they had agreed to take him based on seeing only one out of the eleven appendices my DS has to his Statement!!! (This is now the subject of a formal investigation by LGO.) I also asked the teacher in charge about their experience of dyslexia, pupils with dyslexia etc etc - of course she answered "none". So that all went in my report. The one thing I forgot to do was to get her to sign it - which she probably wouldn't have done anyway. But that didn't matter because the judge asked her during the hearing if she had answered those questions on my report herself, and of course she said "yes". So both the FOI and the teacher at the LA showed the school couldn't meet need so why were we at Tribunal?
One of my EPs also went into LA school and assessed it to see if it could meet need - it couldn't
So we just built a very strong case that the LA school couldn't meet need. I have heard of people who also sent in an EP to the indie school to determine if it could meet need - but we didn't have to do that
It was a very painstaking lengthy process as my LA do not believe dyslexia exists (they told the judge that) and they are a very nasty vindictive unprofessional bunch of people.
But we got there in the end. Tribunal judge ruled that LA school could not meet need because a) they could not meet the severity of his dyslexia and b) he is so complex with other needs that the LA school would not be able meet all those needs anyway. We also built the case that even if school could meet need, LA school would cost more then indie school. But because LA school couldn't meet need, costs didn't even come into it.
You need hard evidence for this but it can be done. Must LAs will fight you to the bitter death if you want indie ss, so you'll probably end up at Tribunal.
Hth and good luck