I've been through this.
You should have a multi-agency transition planning meeting in Year 9 to draw up a Transition Plan for your son's transition to post-16 education and into adulthood (to age 25).
The repsonsibility for calling this meeting falls to the Head Teacher and she should invite evryone who is involved with your son i.e. you, school, medical professionals, social services if appropraite, Connexions (if they still operate in your area) and the Ed Pysch.
Together you are supposed to identify what support your son needs (not just educationally) post-16 and how that support will be delivered i.e. educational placement, etc. The decisions hould be recorded in the Transition Plan and actions then start being taken to implemnet the Plan.
The LAand Connexions should find him a suitable post-16 placement. That usually means that school will attempt to wash their hands of him as they'll claim that they only have responsibility for children while still in statutory education i.e. their responsibility ends at 16. That's not true.
If a suitable school can be found (almost certainly by you) then you can ask the LA to amend the Statement to show that school - they will almost certainly refuse as it will almost certainly be an independant placemnet that will cost them a fortune. So you'll probably end up at SENDIST when the LA try to cease the Statement because your child is now 16.
So the LA will try to pass the problem to Connexions, who will be looking for local FE place (as you already know).
To get special residential placemnet in FE College you have to prove that there is no local FE College that could provide your child with the level of support he requires. You also have to prove that he needs 24/7 care - i.e. that he also has social needs. That can be because of social isolation becsue he has ASD or needs continous placemnet to be actively taulght social and communication skills in a 24 hour setting and also needs to gain indepence skills he could not obtain at home.
Connexions can assist in making that case. You'll need lots of evidence and reports etc. Make sure you get a Connexions Specialist Advisor as most Connexions PAs have no SN training and don't know how to obtain a placemnet - or even that such a thing exists!
You're probably looking for something like Farleigh FE College, Frome. They board there and come home alternate weekends. Fraleigh staff take them to local FE Colleges each day and provide 1:1 support within the local FE College if necessary. Young people at Farleigh can stude academic or vocational courses, have social skills training, independence training, art therapy etc etc etc.
But it's very very expensive so you'll have a fight to obtain this.
Best thing is to do your own research for suitable placemnets. Don't but the LA's lie that there is no money etc etec. Ignore that rubbish.
Many of the possible schools / FE Specialist Colleges will have Open Days at this time of year. If you ring them they send you and invitation.
You can contact the schools / college directly (I did). They also know how to secure funding and can help you kae that application.
Best wishes