You can use the religious aspect but not by saying how you meet the admission criteria. That simply governs your place on the waiting list. It does not give the appeal panel grounds to admit. However, you can still use it by talking about, for example, the school's ethos and how that would be beneficial for your son. Similarly you can talk about their pastoral care if that is better than that on offer from his current school.
If he does a lot of swimming, is a talented swimmer or wants to improve his swimming you can still use the swimming pool.
I would ignore the education office's advice that other schools with space can offer everything. That is almost certainly not true. This school may offer some subjects that are not available at other schools. It may have particular specialist provision in some subjects. And it almost certainly has different extra-curricular activities. Look at lunchtime and after school clubs, sports teams, musical activities and so on. Ask the school rather than the education office.
The case to refuse sounds fairly standard from the summary you give. If they are over capacity for the school in total the building size is relevant regardless of the fact that there are spaces in other years. However, if they are under capacity it is not. It is impossible to say more about their classroom size argument without more information but it is unlikely to be enough to say they really can't cope with an additional pupil. Dilution of teaching is a standard argument and in general won't be given a huge amount of weight. It doesn't sound like a huge mountain to get over but it will depend on you getting a strong enough case together.
It would be worth finding out if they have had more pupils in this year previously. If they have that is an argument in your favour - they have coped with X number of pupils before so they can cope with more now.
Note that the school (and the LA if it is the admission authority) must answer any questions you ask to help prepare for your appeal, so it should be easy to find out anything you need.