The CTA is a strange beast poorly understood. To avail of CTA rights you have to demonstrate you are a citizen of the CTA, thats always been the challenge. Under 16 doesn't need photo ID but must demonstrate they are entitled to CTA rights. As above carrying the birth certificate is the best/only way.

There is no clear documentation as to the requirements in this case, but not to have any documentation at all would raise questions.

So I'd suggest the following before going in all heavy, you need to ask the UK Border Force what is needed and get that in writing. If they say in writing that nothing is required then you proceed with a EU261 claim against Aer Lingus (the operating airline). If the claim is refused its the Irish Aviation Authority as the competent authority (Irish airline, departing Ireland)