You should have been offered a place /places in a school even if it is not the school you want. The general advice is to accept this place / places because the Local authority will not offer you another one and at least your children will have a school place. If you are lucky the places offered wil be in the same school, if you are unlucky they won't.
Ask to be put on the waiting lists for your preferred schools, you never know, a place, or preferably places, could come up.
It is a standard response for a school to say they cannot accommodate extra children above their allocated numbers. The purpose of your appeal is to demonstrate that your children's need for a school place there is greater than the school's response that they cannot accommodate them. An extra room that doesn't seem to be a classroom won't work, the room will be used for something, group work, social skills, library, resource room. The school is not going to turn it into an extra classroom and pay for a teacher for one child.
By all means use continuity as an argument for your appeal, but I am not sure how effective it will be. Have you made contact with the local church and priest, introduced yourself to the church community? Are there other curriculum or social advantages to the school. For example do they run a maths club and your child is mathematically gifted, do they offer additional music classes or instrumental teaching that your child can continue with.
I imagine that the school's argument will be that they are no better or worse at settling new children into school than any other local school and that you can make local connections through the parish which will support your children's social and faith development.
Be warned that any appeal if your child is in Reception, Year 1or Year 2 will fail on the grounds that if the school is at its maximum capacity they will not admit an extra child . If you can prove your case for children in Years 3 to 6 then there is a chance that the panel could make the school give you a place and go over their number, but this is assuming they haven't already done so with previous appeals.