You say PAN is 23, the LA push it up to 25/26. That rings alarm bells for me. If they are regularly admitting beyond PAN they ought to change PAN. However, if the extra pupils are due to successful appeals there is no problem with that, although it does suggest that the appeal panels think PAN has been set too low.
The figures they have given for net capacity are actually the caculated net capacity. This is based on the number and size of rooms in the school. The lower figure is always 90% of the higher figure. The actual net capacity should be set somewhere between those two figures.
PAN for a primary school is usually the net capacity divided by 7, although they can go for a higher or lower figure. If they have set the net capacity at the bottom of the range (189) that suggest a PAN of 27 whereas at the top of the range PAN would be 30. A PAN of 23 would only give 161 pupils if every year was full, well below the minimum calculated capacity. So there is a clear argument that the PAN is artificially low and the net capacity indicates that they can handle more pupils than have been admitted to date. Definitely worth highlighting this.
I can't comment on the letters beyond saying that the panel should ignore anything from the preferred school. The school is not allowed to support your appeal. It shouldn't do any harm to submit them, though.
As mummytime says, anything the preferred school offers that would be particularly beneficial for your son is worth mentioning.
And I agree with tiggytape that you should remember you are appealing for this school, not against other schools. Criticising the schools you don't want can put the panel's backs up. After all, one of them may have a child or grandchild at one of these schools.
The logistical issues, travel and childcare won't win your appeal. As tiggytape says, the appeal panel cannot consider these factors.
I'm not sure how strong your case is, but the discrepancy between PAN and the calculated capacity, if your figures are correct, suggest that the case to refuse admission is weak. The LA should have included the actual net capacity in their case. If they have not done so, make sure you ask them for that figure during the hearing. You should also point out the figures given for the calculated capacity last year and ask them to confirm that these are correct. Assuming they do confirm these figures and the net capacity is somewhere within the range 189 - 210, state that the capacity points to a PAN of x (where x is the capacity divided by 7) and ask why they have set PAN so far below the indicated figure.