As @TeenDivided says, to win you need to show that the disadvantage to the child from not being admitted outweighs any issues the school will face from having to cope with an additional pupil.
How much emphasis should I place on their asylum seeker status
Difficult to say. This is one of those things that really depends on the panel you get. Some will think this is highly relevant, others less so. You should definitely include it.
Although the church linked to the school isn't the one we attend, lots of children from our church go there so they already know lots of families and children
I wouldn't regard this as a strong point on its own. It might carry more weight if linked to the asylum seeker argument, but don't spend too much time on this.
The Christian ethos of the school is important to the family but is that worth mentioning when the school they currently attend is Catholic
As the child is already at a Catholic school, this doesn't help. I would ignore this or, at most, mention it in passing.
I know the school went over PAN to accommodate Ukranian children
If they have been over PAN, that is definitely relevant. The school is required to answer any questions the parents ask to help prepare their appeal, so they can find out how far over PAN they went and when it happened.
Is there anything else I can include?
Look for things the appeal school offers that are not available at the child's current school and are relevant to the child. For example, if they are interested in music or have musical talent and the appeal school has more extra-curricular musical activities than the current school, you should bring that up.