Nobody can tell you your chances of winning, because the panel has to weigh your case (you want your son to attend this school) against the LEA/school's (they don't want to admit beyond their admissions number) and we don't know how strong, or not, the LEA's case will be.
That said, I think you have a few things here that any panel will give a lot of weight to. First, that he is a young carer and needs a school with excellent pastoral care. If you can find evidence that the appeal school is best for this - it's won an award (say) or Ofsted drew attention to it - that will be very helpful. (Otherwise, the LEA is likely to argue that all schools offer pastoral support). Second, his sporting achievements. If you can submit evidence of the appeal school's superior facilities and record here - they are county track and field champions, say - that too will help. Does the appeal school have any special expertise or support for dyslexia or dyspraxia, beyond what's standard in secondary schools? If so, that's a third area where you can argue that this is the best school for meeting your son's needs, which is what any successful appeal usually hinges on.
You don't need to go to town on Ofsted ratings, as the poor rating of the neighbourhood school isn't really the issue. You need to identify positive benefits for your son in attending the appeal school, not reasons for not selecting the other school.
Dig out the figures for how many pupils there are in other year groups, or have been in Y10 in recent years. What you're looking for is evidence that they have exceeded their usual limit in recent years and (presumably) the sky hasn't fallen in. That enables you to argue that, despite its inevitable protestations, the school would cope if your son was admitted.
The letters from professionals you have gathered sound very useful, but make sure they're not just generalised statements about his needs but state plainly that in the professionals' opinion, the appeal school is the best one to meet those needs.