From the point where an incorrect documentation issue was raised and you still wish to travel alone.

I traveled with my wife SIN - LHR back in 2017, she forgot her FLR card but had a copy of it. At first, check in was refused for us as a group (Me BC, wife ILR and daughter BC), WHICH WAS THE POINT where I asked if I can still fly back to the UK alone. The answer the agent gave was yes, effectively splitting our joint booking.

The issue got resolved when the agent at SIN phoned UKBA up which gave her clearance to fly as her copy allows them to verify her leave to enter on their system.

Sorry yes, I should have said, that if your state provides a EU nationality card, then you can travel on this. You can’t just travel in any ID such as a driving license though (appreciate some states may combine driving licenses and EU nationality cards- it is the later part that they would be travelling on) UK has no such card and therefore a passport is required.

I would advise against that, unless your solicitor offers a different opinion. This case is likely to fail "merit test" at hearing and repercussions of grossly wasting courts time could be quite harsh.
If you are truly determined to take this case to court of law, please consult a lawyer first.

Do BAs CoC require a toddler (UK/Eire citizen) to have a government-issued photo ID for flights between Ireland and the UK?

That would be the main question for to determine, whether EC261/2004 article 7 compensation + article 8 free rebooking applies.
If the toddler would have been allowed to travel without ID then BA would be on the hook for EC261/2004 payments.

Moreover, I know cases where an airline denies boarding to only a part of a family.
In most cases the family then decided to not travel at all (to avoid a family seperation) -> I'd assume that an airline would always argue that it owes only EC261/2004 compensation to the ones who were actually denied boarding.

I am not the one insisting BA check in agent is at fault here.

Sorry, should have made clear that was not meant to question your judgment (I am all for watching a good fight from the sidelines) but rather OP's. Over hundreds of years British and European courts have mastered an art of protecting themselves from abusing the system and frivolous litigation. Wish the US followed the suit.

The CTA provisions only apply to citizens of the UK and Ireland. The OP did not provide any evidence the child is a British/Irish citizen and consequently the child could not enjoy the CTA travel provisions. All passengers who are not British or Irish must provide a passport for travel to/from the UK and Ireland.

Is it is a bit of a circular argument though? UK/ROI citizens don't need documentation to travel, but you need documentation to proven you don't need documentation?

IMHO, the OP has no claim here. Any halfway competent judge would point out that there was no way he was getting on the plane and abandon his son at the airport.

With that said, it is situations like this that boggle my mind from an airline perspective. You are sitting there as a check-in agent facing a traveler who has clearly made an innocent mistake. He isn't trying to game the system (at least at this point in the narrative). Why is it so difficult to simply confirm him on the next flight? Even if they fare rules say otherwise, this is what exceptions are made for.

JS

UK/ROi citizens need ID as approved by the airline to fly or to board a ferry. They don’t need to produce it to cross the ROI/NI boarder and they don’t necessarily need a passport, just ID that confirms identity and nationality.

The rules are clear. In my experience, the behaviour you exhibit towards the staff will largely determine how much they are then able to bend the rules, which they do not have to.

It’s a bit like police officers who talk about traffic stops, if your a dick to them for stopping them, it is clear that letting you go on your way s unlikely to prevent similar future behaviour, whereas if your not (insert suitable disparaging description), they can undertake speed awareness education and it is more likely to have an effect.

As I said previously, I was not there to witness the encounter, but it is clear from the OPs description, that relationships clearly broke down.

In practice, yes. But in theory, to be enforceable this would need to be in the Conditions of Carriage, and it's not clear if there's a requirement therein to possess documents over and above what the immigration authorities might require.

And it is in the COC of every airline/ferry operator.

I bet the wife is relieved it's being dropped! That is probably worth more than any possible compensation!

I'm being slow and cannot find it... Sorry.

For BA, see section 13 including section 13f.