In my experience, all third-party bookings show up as "Smart" in the App.

You're travelling on a 125-issued ticket, so your ticket does not actually map to any of the Aer Lingus fare families; all third-party tickets (issued by BA, UA, AC, or any other airline) exist as a sort of bog-standard default economy class, including 1 checked bag, outside of the Aer Lingus fare family ecosystem. Perhaps things have changed since April (when a greater degree of integration commenced, in so far as Tier Point earning was "switched on" for BA customers travelling on EI, and Tier Credit earning was simultaneously enabled for EI customers travelling on BA), but in my experience all third-party tickets issued for Aer Lingus flights never actually book into "Aer Space", and I do not believe such tickets ever qualify for same-day standby.

That said, on LHR-DUB and DUB-LHR, I have frequently been proactively moved to earlier flights - without status or holding an Aer Space booking - so it's not unheard of for them to move pax even if not strictly allowed by the fare rules. You can ask, but you shouldn't expect it to be available always.