If your son has an EHCP the process for applying for primary is different and the criteria for school selection are different to a standard reception class application.
The key issue is whether the school can meet his needs (as described in the ehcp) and whether it is the closest school to you that can meet his needs. Out of borough schools would normally only be considered if there isn’t an in-borough school that can meet his needs (not uncommon, but you would have to prove this using evidence from professional reports and his current setting).
So my advice would be to start looking early, visit all the schools you think might be suitable (especially the Lambeth schools the borough may think are suitable, as you would have to prove why they wouldn’t be). Talk to other parents and try to find out what schools are like from the inside.
Also, bear in mind the journey and that he (and whoever takes him to school) will be doing it twice a day for years, whatever the weather. I have a child with an ehcp placed out of borough, and it has had a bigger impact on social relationships, being part of the community etc than I had anticipated, so that’s something to think about too.
If you need some independent advice, the charity SOSSEN has drop in clinics that can be very helpful in cutting through the conflicting answers you may be getting from schools and borough advisors with straightforward advice on the law.