The fact you only listed one school meant you only had one shot at avoiding the Catholic school. From your posts, it is the nearest school for which your child is eligible, so you were always likely to get it if you didn't get your first preference. However, there is no use crying over spilt milk and some of the posts on this thread do seem to be putting the boot in unnecessarily.
What can you do now? Well, the first thing you should do is find out what schools in your area have places available. You won't get free transport but, given how strongly you feel about this, I'm sure you would find taking your child to a school some distance away preferable to having them attend the Catholic school.
Secondly, you should apply for any schools you would prefer even if they don't have places available. You will be turned down, but you can then appeal for those schools as well as your preferred school.
As others have said, your religious objections will not carry any weight at appeal. To win an appeal, you need to find things that the appeal school offers that are missing from the Catholic school and are particularly relevant to your child - subjects, extra-curricular activities, facilities, etc. You can use those to show that your child will be disadvantaged if they don't go to the appeal school.