I've noticed on ba.com:

LHR-DUB only shows the BD operated flights.

(some other country)-DUB shows only the BA coded EI flights from LHR and not the BD ones.

(UK regional)-DUB doesn't return anything, even if the route could be entirely operated by BD (e.g. EDI-DUB).

So I don't think this will affect traffic on the BA/EI codeshare because the people who can book the codeshare aren't offered the BD option.