I find your post quite confusing to be honest.
Firstly, who has said he is bottom of the class? That is not information that any teacher with tact would ever give out to parents. Of course they should share if he's behind, but did they really say he's bottom of the class or is this just something you have assumed based on the report that he's behind in some areas?
If they say he is expected in maths, trust them. They know him. This other school doesn't. It's only natural he may have underperformed with people he's only just met. Him being considered expected at his current school but behind in another school that have assessed him on some sort of entry test does not mean the other school are 'ahead' in their maths curriculum. When children join a new class, many of them will underperform a bit until they find their feet. That is natural.
If the school have been running interventions with him they definitely do care. Things will click when he's ready in reading and writing. It's very normal for some children to not meet the expected level by the end of Reception, especially summer borns, if they are simply not ready yet. You acknowledge yourself that he's immature and will likely catch up by 7. I'm sure that will be the case.
The fact his current school's interventions haven't worked yet doesn't necessarily mean they're not good teachers. These things take time and patience. They're not going to work overnight. Your comment that you, despite not being a qualified teacher, are probably a better teacher than them is extremely disrespectful. I'd like to stand you in front of a class and see how well you do. Teaching in a school is incredibly different to working 1-1 at home with your child.
At the end of the day, if you strongly feel you wish to move your child to this other school it is your prerogative to do so, but I do think you're being incredibly harsh and rude about his current school when there is no real evidence from what you've said that they're doing a poor job. I understand you're disappointed your son is struggling in some areas but they are putting things in place to help. Miracles don't happen over night.
Many people make the same mistake as you in believing a smaller school means extra support. Smaller school have less staff and resources. Levels of support are roughly equal in all schools, regardless of size. It sounds to me like you're expecting too much and are looking for someone to blame for your son being behind, rather than accepting there are factors behind it. Not all children will be working at the expected level. There's no shame in having a child who isn't. From what you've said, his current school are working to help him with the areas he struggles in. What more do you want? They can't simply make him expected just because you want him to be.