You need BA silver or better to access the EI T2 lounge, regardless of class of travel, codeshare or ticket issuer.
Valid LHR to DUB/SNN/ORK (and DUB/SNN/ORK-LHR/LGW) on EI metal only
The whole mess dates from EI leaving OW, EI had been given the OW monopoly on IE-LHR and bought a fleet of A321's to operate, when EI left OW, BA needed a codeshare and a deal to access the lounge, EI FF get a much more limited reciprocal deal in T3/T5