I talk from bitter experience of having had a DC in a mainstream Ofsted outstanding indie school for 4 years until I had to pull him out to home ed him, before he had a nervous breakdown because of their appalling attempts at "fixing" his SENs. Never, ever again... (This was pre-diagnosis/pre-Statement with full fees & SEN support paid for by myself.)
I would strongly recommend that you look around the school and talk to other parents - including parents who have a child with a similar profile to yours. Don't be taken in by glossy open-days/brochures/bumpf - the existing parents of SEN children will be the ones that can give you a truthful insight. Look at the qualifications of the teachers/therapists. You may be surprised at the lack of qualifications for mainstream subject teachers.
Talk to the SEN teachers and get a feeling for their teaching/therapy methods. Don't be taken in by the small class-size - my DS was in a class size of 15 and they still couldn't cope with his SENs, and the class size was still too big for him.
With regards to fees, it looks as though there's two things going on here a) school bursary (or bursaries) and b) a Statement whereby the LA will pay for the fees of the school.
I can't comment on a) - that's the school's own individual policy. You could talk to the Bursar and they should be able to help you and tell you what options are open to you. Usually bursaries are not for the full amount, it'll be a percentage of the yearly fees - and you will have to meet their criteria.
As for a Statement of Educational Needs, and then the local authority paying the fees... You will be in for one hell of a fight to get this. The LA may agree that your child needs a Statement, but it is highly unlikely that they'll name an indie school without you fighting tooth and nail for it. It is more likely that the LA will name a state mainstream school and, if you are set on an indie school, you will then have to prove that your DC's needs can't be meet at the state school they name. It can be done - it has been by others on this board - but you will be in for a fight.
Indie schools tend to use independent EPs, OTs and SALTs who are likely to have their own independent practice so will be ethically bound by their own "bodies". In my experience, they will be excellent for providing the direct therapy but won't be used to writing reports that can be used to persuade a LA to undertake statutory assessments.
Many indie schools will support children to get a Statement but once the child gets the Statement, some schools will then panic and ask DC to leave (this is what happened to other children at my DC's old indie mainstream school). They do have to put in force anything in a Statement, but you (or the local authority if your DC has a Statement with the indie school named in part 4) will have to meet the additional therapy fees/TA fees/extra fees etc.
However, having said all of that, there are some indie mainstream schools that are very very good with SEN children. But after my appalling experience, I would be very very wary.