Sorry, with my careless speedreading, I didn't catch that the child was already in Y6
I agree that the normal assessment season is coming to an end. However, this is only important for schools that fill all their places. If the school is undersubscribed, you can apply any time. If they want your son, they will make him an offer and may be able to offer some financial assistance too.
I wouldn't get too hung up on whether its a scholarship or bursary. Scholarships are more public, whereas bursaries are confidential. There may be a fixed number of scholarships offered, whereas the situation with bursaries is more clandestine, unless the school has a huge endowment.
The only way you will know is to ask.
As for 13+ entry, these are usually for pupils coming from prep schools and taking the Common Entrance. The school may have its own alternative to CE. The syllabus is virtually the same as KS3, if taking French at lower level. Latin is optional. The exams are usually taken in prep schools, the week after summer half-term in Year 8. This is preceeded by a a VR/NVR test in late Year 6/early Year 7, which is where the real selection takes place - CE is a formality, tbh.
By their very nature, independent schools set their own admissions procedures. There are no hard and fast rules, and most schools can be flexible and try to accommodate your individual needs. What you need to do is to ask the school.