The relationship between EI and BA is complicated.
EI Arrivals Lounge - Access only to those arriving in J. No guests.
EI Lounge - BA gold and silver if travelling on EI metal to LHR or LGW - although any EI route with a BA codeshare may, unofficially, gain access, depending on the door. The passenger doesn't have to be on a BA flight number.
BA use the DAA lounge at Dublin, - standard entry rules. For their own flights rather than the EI lounge, so you would have had access there.