as adrianlondon asked, what evidence do you have that it is allowed? you mentioned you triple checked, where did you triple check? for an IDB claim you have to show that you were correct and the denial was erroneous.

EDIT: there seem to be two issues
- can you transit the UK on a trip to Ireland if remaining airside in T5 without an ETA
- were you able to cross UK immigration (you would do so since you are entering the common travel areas during your connection at LHR) with an EU identity card and without a passport if connecting on to Ireland

on the second point you mentioned you are not entering the UK. however i think you may not appreciate the UK and Ireland are in a common travel area and when connecting in the UK for onward travel to Ireland, even if staying airside, you must pass through UK immigration.