By 'multi-city itinerary," do you mean that you're planning to stop over in DUB? If so, you can try booking each leg separately; if not, you do not have a multi-city itinerary, you just have a connection, and you should do an award search for LIS-SEA: if LIS-DUB-SEA is available as a single award itinerary, it should show up in the results.
AAdvantage benefits on Aer Lingus here:
https://www.aa.com/i18n/travel-info/...aer-lingus.jsp