If your daughter would be going into Reception, Y1 or Y2, there is a good chance that this would be an infant class size appeal, in which case I'm afraid it is highly unlikely you would succeed. To win, you would need to show that a mistake has been made in the normal admissions process and that your daughter should have been given a place at the appeal school. As the schools are full, it is unlikely that there has been a mistake. The only real argument for a mistake would be if your daughter could have been treated as an excepted child, which means she wouldn't count towards the infant class size limit. Unfortunately, you haven't said anything that indicates your daughter could have been excepted.
If it is not an infant class size appeal (i.e. it is for Y3 or later, or the classes in the school have less than 30 pupils per teacher) you have a better chance, but the arguments you have advanced will not get you anywhere.
Going to a Catholic school does not mean your daughter will be brought up in the Catholic faith. Some Catholic schools are less religious than some non-faith schools, and you have the right to remove your daughter from assemblies and RE. Many non-Catholics go to Catholic schools.
It is understandable that your daughter wants to go to a school where she already has friends, but friendships are pretty fluid at this age. Unless you had evidence from a medical professional that your daughter has a much stronger need to stay with friends than other children of her age, this argument won't fly.
The preferred school better suiting the way your daughter learns is rather vague and, again, unlikely to fly at appeal without expert evidence.
You have nothing to lose by appealing but, if you do, you need to be realistic about your chances.