I haven't read the full thread, but I agree with @MargaretThursday that your son's academic ability is not relevant to the appeal. Looking at the points from your first post:
It offers strong pastoral care, which is vital given his anxiety and emotional needs.
That's a decent argument, especially if you can convince the panel that he will need this. Evidence from a medical professional on this point could help.
It provides continuity of education, including subjects he is currently studying and wants to take at GCSE (such as Spanish and Design & Technology). In addition it closely follows similar ethos, academics and oppprtunities to his current school.
As he is going into Y9, the availability of subjects for GCSE is relevant. If these subjects are not available at other schools that you might be offered when you return to the UK, this is a strong argument. Ethos, etc., are not as strong but still worth raising.
Being only 0.8mils away, It is in close proximity to home and the school that will be attended by his younger sibling, which helps with family logistics and support.
This is unlikely to carry much weight. I understand your argument about his unfamiliarity with the surroundings, etc., but the panel is likely to view this as a problem for you, rather than for him. By all means raise it, but don't spend too much time on it as it isn't your strongest argument.
The short distance would reduce his exposure to stressful travel, which could worsen his anxiety and asthma.
If you have medical evidence to support this, this is a better argument. By supporting evidence, I don't mean just letters saying he has anxiety and asthma. You need a medical professional to say that, in their professional opinion, your son needs to go to a nearby school for these reasons.
As a single parent and carer, I would struggle to manage school runs if he were placed further away.
This won't carry any weight at all. It is definitely a problem from you, not for your son.
Not attending this school risks worsening his anxiety and asthma symptoms.
If you have evidence from medical professionals supporting this, it will be a strong point for you. If you don't, the panel is unlikely to give it much weight.
What other types of evidence or documentation would strengthen my appeal?
I don't think you need anything else apart from the things you mention and the things I've mentioned above.
Does this sound like a strong enough case based on medical and social grounds?
Possibly, but it will depend on the strength of the school's case to refuse admission and may also depend on the appeal panel you get.
Any tips on presenting the appeal, especially addressing my son’s current emotional state and health concerns, independent travel or lack of prior UK schooling?
As this is an in-year appeal, the panel won't be under as much time pressure as for normal admission appeals. That gives you a bit more latitude in presenting your case. However, don't make it too long and concentrate on your strongest points. Try to look for weaknesses in the school's case and come up with questions for the school's representative that will highlight them. Be ready with a short summary of your case as, after you have presented your case and the panel and school have asked you questions, both the school and yourself will be asked to sum up. And try to relax - easier said than done, I know.
Our LA won’t let us apply for a place until we are back in the UK end of June
As that is only a few weeks away, it doesn't really matter but your LA is not following government guidance. See School applications for foreign national children and children resident outside England - GOV.UK and note that it says that an admission authority cannot require the applicant to have a home address in the UK before processing an application.
How do I find out their current PAN and cohort numbers? I can see them for 2023-24 online but not for the current year. Would this help my case if I can show that the school has admitted higher numbers in the other year groups which supports they could cope with an additional student?
These things normally form part of the school's case, but if you want to make sure you can just ask them. They are required to answer any reasonable question you ask to help you prepare for your appeal. And yes, if they have been over PAN before, that can help as it shows they can cope with an additional pupil in the year. However, if several years are over PAN that may hinder your appeal as the panel may think the school is already overfull.