There's no nonstop path connecting Louisiana's most idiosyncratic city to Spain's Basque coast, which means every itinerary in business class from New Orleans to Bilbao involves a strategic layover — and the routing you choose matters more than the fare alone. The most reliable option runs through Madrid on Iberia, which operates modern A330s and A350s fitted with fully lie-flat business seats in a 1-2-1 configuration on the Airbus 350, giving MSY travelers a seamless one-stop connection into Bilbao via Iberia's regional network. British Airways offers an alternative through London Heathrow, where its Club Suite — a proper door-equipped, fully flat seat — has become the benchmark for transatlantic business class, though you'll need a subsequent short-haul hop on to Bilbao from Terminal 5. Lufthansa routes through Frankfurt or Munich, and its newer long-haul aircraft are gradually introducing the Allegris cabin, though not all MSY-bound equipment has been retrofitted yet, so it's worth checking aircraft type before booking.
Having flown all three corridors repeatedly, my strongest recommendation is the Iberia-via-Madrid routing whenever schedules align, because Madrid–Bilbao is a short, frequent shuttle-style hop that minimizes connection stress after an overnight transatlantic flight, and Iberia's Madrid hub allows a relatively relaxed transfer without the security gauntlet you'll face at Heathrow. An insider tip few travelers know: Iberia's business class fares into secondary Spanish cities like Bilbao are often priced identically to Madrid-only tickets, so there's little reason not to book straight through rather than buying a separate domestic leg.
Deal-wise, this route favors the shoulder seasons — January through March and again in October and November — when Basque Country's wine harvest and Guggenheim exhibition calendars quiet down and transatlantic business fares soften noticeably. Given how much these fares fluctuate week to week across Iberia, British Airways, and Lufthansa, setting up automated fare alerts is the smartest way to catch a premium seat in this corridor before it disappears.