Your LA should send you a list of special schools, including residential and independent schools, on request. There are a few good books which list special schools. I have used the following:
Gabbitas guide
Good Schools Guide
Which School
I think the Gabbitas guide is the most comprehensive, while the GSG gives a bit more information based on actual school visits. There are also ways to search online for independent special schools - e.g. ISBI
You have the right to ask for an independent special school but these will usually cost more than a state special school. In order to get your LA to agree funding, you need to prove that the state special schools cannot meet your child's needs and the independent special schools can.
The LA only has to provide the school which is most adequate for your child's needs, not the absolute best that is available. Your child sounds like he has quite complex needs which is useful - many state schools will not be able to cater for all of his learning/social needs.
Most parents looking for independent special schools end up commissioning independent reports, from educational psychologist and therapist (SALT/OT etc), at a cost of £500+ each.
An independent report is far more likely to make recommendations which are tailored entirely to your child's needs and not based on funding priorities. It's expensive, but worth it in the long term - special residential school fees can run to hundreds of thousands of pounds per year.
To start with, I would make a list of the schools which say they can meet your son's primary need (as stated on his statement). Then visit their website and Ofsted reports to narrow it down. Then, if they still seem suitable, start ringing around (you may have to wait until term starts again) and have a chat to see if they think they might be able to meet your son's needs.
You can send your son's statement to them yourself - don't wait for the LA to do it for you. They will not be proactive in seeking an independent placement and you do have to take the lead in doing this.
Good luck - it is a long slog but well worth it. I have secured an independent residential placement for my son with complex needs, and having visited over 30 schools in the past few years, I really believe that the independent sector provides the best provision for SN.