I work as a school admissions officer in a local authority and I also do the paperwork and present appeals for the schools that the LA is admission authority for.
I would say that one or two appeals are successful each application round for our schools- almost exclusively Y7 rather than Reception as the bar is so much higher to breach infant class size legislation.
I always advise people to appeal - we lay out what they need to do and we are realistic in warning people that all schools can support mild ASD, that travel/work/childcare arrangements will not be a good reason etc but that is not to say that the appeal decision is predetermined just that what seems like a big issue to your family is unlikely to be something that the panel haven't heard 100 times before and that generally boils down to parental preference rather than a need for an extra place in the school.
One year when I was shadowing the previous job holder, the panel decided that the school's statement was not strong enough and there was nothing to suggest that they couldn't handle an extra child in each Y7 class - they allowed all the appeals even the one where the Mum said that she really wanted X Catholic School down the road but thought she may as well appeal for this one too and the ones where the appellant didn't show up.
A year or so ago similar happened with a local Academy who arrange their own appeals - the Head presented and was asked if there were any spaces in Y7 and she said 'yes' - what she meant was 'yes, but there is a waiting list and the places will be offered to the children at the top of the list' But because she didn't say that the panel allowed all the appeals - regardless of their reasons for the appeal.
These things happen - in our case - where a mistake has been found, we will make sure the place is offered and it never gets to appeal (mistakes are very rare and it is normally something like a school not recognising a sibling so priority is not applied).
With the ones like OPs in my experience panel members are kind - they become panel members because they have an interest and because they care - they want the system to be fair and for people to have a chance.
The emotional ones like bullying etc are the one or two that can be successful - but it very much depends on how the appellant comes across, but they want to help in a way the admission arrangements can't - to use their power to make the right decision for a family who won't get the help anywhere else
Show that DC has lost weight, is anxious, sacred upset etc - gather as many statements from professionals as you can to support this.
Don't slag off the school you have been offered - talk to the school you have been offered to see what they will do about the issues so you can explain to the panel you have spoken to them to see if it can work and to reassure DC but that you can't see that it will work.