Absolutely no problem do this
You can check in online with the BA website for the BA flight no problem regardless of checking in for the EI segment, works for me every time.
1. Go to BA website NOT the app and try to check in for the EI leg and they will give you the PNR ref for EI and direct you to EI's site or you can be successful from time to time with the BA website doing OLCI for EI if a seat is already assigned but this will only work with a printed PDF boarding card.
Neither EI or BA will offer seat selection choice it will issue a random seat unless there is one assigned. If you manually paid for a seat then you may not be able to check in online if you did so less than 72 hours before departure.
2. As above but if you have the EI PNR skip the trying to locate it on the BA site and go direct to EI
If you check in in Dublin in person the staff will handle any passport/visa check requirements to avoid any delay at flight connections in LHR. Staff at check in will typically prioritise connecting passengers with seats towards the front (it used to be the back in the T1 days, but its front now in T2)
Always check in with the operating carrier for each leg, EI's airport IT in Dublin is extremely good at issuing boarding cards for all manner of airlines even some they don't list
As a BA gold flying DUB-LHR you have access to the EI lounge, no guest