I think the core question is
"Is the current school meeting his individual needs well? Are they enabling him to make good progress?"
Another important one is
"Do you, and do the school, believe that after a second run through reception, he will be in a position to progress into Year 1 closer to the average of the cohort, and then progress at the rate of that cohort?"
Leaving aside the state / private discussion, I think the question that always needs to be asked about repeating a year is whether it is considered likely to 'fix' the problem, or whether it simply postpones it. For some children e.g. very premature or for whatever reason not ready for school at 4, deferring a year enables them to progress normally within their new cohort. For others, repeating the year still leaves them at the bottom of the new cohort and progressing at a slower rate than their peers.
Returning to the specific situation, if you think that the school is absolutely the best setting for him, and is meeting his SEN needs and general needs very well - and will continue to do so - then repeating a year is worth a try. You may want to ask to speak to parents of children with SEN higher up the school, and also question the head and SENCo in great depth about any lower ability or SEN children who leave the school other than at 11, because 'managing out' is appallingly common (I speak as a teacher in a school which by virtue of its location 'catches' managed-out children - I would go as far as to say ALL children transferring from local privates to state during the primary years have SEN or are of significantly lower ability ).
Then I would call a range of post-11 schools - from those that specialise in children with SEN - if you have any private schools of that type locally - through 'averagely non-selective' to 'standard destinations from your DS's school' and discus their policy and experience of a) out of year children and b) children with ASD / ADHD.
I don't know what your circumstances are, and why state schools are not possible. Remember that if your DS has one or more diagnoses, and his needs are significant, he may be able to get an EHCP, which makes it possible to name the school he attends - so if your reason for using the private sector is to do with local school options, children with EHCPs do not always have to attend their 'catchment / most local' schools, if schools elsewhere can meet their needs better.