Delta, Air France, KLM, British Airways and various other airlines have round-trip flights from Austin or San Antonio (AUS/SAT) to Dublin (DUB) for $352-$419 with a connection. Flights June to March. 7-day minimum stay required.
Some dates have longer connections that others, so you may want to try several dates to see which ones have the best routing.
COVID travel restrictions are still in place but this may be worthwhile for the dates later this year and next year.