A BA Silver card (and it has to be issued by BA - any of the oneworld equivalents, such as your CX card, do not count) would actually grant you access to any of Aer Lingus's Gold Circle lounges when flying between Heathrow and Ireland on any Aer Lingus service which carries a BA codeshare number - even when the ticket on which you are travelling has nothing directly to do with BA or an onward BA journey (i.e when booked directly from EI or a third party and purchased with an EI flight number).

Aer Lingus doesn't have its own lounge in Gatwick, though, and I really don't think that the Number One lounge there (which is the one EI uses, I believe) would honour a BA Silver card for access. And a BA Silver card would not have gotten you into the Gold Circle lounge in Dublin ahead of an Aer Lingus flight to Chicago - the only time you would have access in Dublin is ahead of an Aer Lingus flight to Heathrow.