Was going to say cerebra for private SALT
You can request an early review of the statement or apply for a statutory reassessment on the basis the current statement is not meeting need.
You need SLT quantified and specified in the statement eg how often they come, what they do and how long each time. If NHS then say they can't do whats needed then LA must by law buy it in elsewhere. So you probably need the wording in the statement changed and also provision for training up of school staff.
You can get a private SLT to see your child and write a mini report for £60-120 for 1-2 hours. That might be enough to show you mean business.
If SLT is in the statement but not being implemented then you need to do a formal complaint to LA and go to Local Govt Ombudsman - it is the LA's duty to make sure what is in the statement happens - see IPSEA common problems page on website for a draft letter - or you can consider a judicial review if LA don't sort it.
Also talk to your school - can they afford to fund training themselves eg ICAN courses etc? Most rely on LA courses which are not usually anywhere near enough but there are other options out there if school will pay themselves.
Are you claiming everything eg DLA which might help towards private SLT?
If you suspect that even fighting for SLT is a waste of time as more SLT will just mean more crap SLT (this is the situation we are in) then you should probably look at SS, units, indep SS etc and ask for a change to the type of school. If your LA is an SEN Pathfinder then ask for direct payments for SLT
I know exactly what you mean about not knowing what you are fighting for, if you suspect that something is not going to be worth having then don't waste your time - find something that is worth fighting for.
Is a dual placement a possibility or linking your school with a local SS and staff coming in to work with staff at your school?
You are stuck in the classic roundabout of professionals talking about what they will do / whats needed but not actually doing anything - this is what we had and was why we took the LA to tribunal to get private provision who don't spend £££ looking like they are terribly busy while actually achieving nothing.