I am assuming this in an in year transfer rather than a phase transfer at the end of infant school, for example.
At Annual review (or early review if you/school call one) you can assert that his current school can't meet needs. If this IS the case and the school agree, the LA will have to consult with other schools, including the private school you would like.
IF the private school (including transport - which you can offer to do for the private but not for another school) is the cheapest school that can meet needs, the LA should name it. If they refuse then you go to appeal and have a good chance the tribunal will name it. You can choose to pay privately in the mean time (good evidence it CAN meet needs) BUT run the risk that the tribunal don't name it (the LA could easily find a SEN placement that can meet needs if mainstream are saying no, or a mainstream that say they can meet needs and it would be down to you to prove these schools are incompatible with your child's needs). You then have to move your child to the school the tribunal name or pay privately for the school and all requested support.
If costs are marginal - ie your choice is more expensive but only by a bit, then the LA may offer to name it as "parental preference", paying fees but not transport and possibly not additional SEN support (if you want THAT paid for then the school have to be very upfront about the full costs for your child to attend ie annual fee of £x and top up fee of £y (which may be itemised) to provide everything in section F not offered to all children in the normal school fee)
If you are in the situation where mainstream can meet needs but you'd prefer private (maybe you'd be paying for private school regardless of your child's SEN needs), you really need to have some discussions with the LA/caseworker. This can be easier at phase transfer as the child has to move schools anyway.
All LAs are different here and some local knowledge would be useful. Its about the money so, if the funding for your child, currently, is £4k placement plus £6k notional + £2K top up, for example, you could well get £2K pa towards therapy or SEN support offered by the private school but this CAN NOT be used towards school fees.
Some parents have been successful with bargaining more than that, actually being given a certain amount pa "to make your own arrangements" which may be more than the top up and then can be used towards fees.
Some parents have been successful in asking for a personal budget to pay for SALT/OT/dyslexia support mentioned in section F out of school but then making their own arrangements for schooling at a private school or home educating. (EOTAS is an option there in section I but only if NO school can meet needs)
So.... can the current school meet needs, in your opinion? If not, I'd recommend planning on a private EP and or SALT and or OT report to demonstrate this if you are wanting the LA to pay for everything, especially if the school aren't on your side here.