British Airways, American Airlines, and Finnair have round-trip flights from Austin (AUS) to Geneva (GVA) for $548-$569 with a connection. Flights in August. Departures on Monday, Tuesday, and Wednesday. 7-day minimum stay required. Limited dates available.
A Carry-on bag and also a personal item is allowed for free on American. You can avoid paying for seat selection by accepting the randomly assigned seat at the gate.