Bilbao's Basque coast rewards travelers willing to route smart, and there is no nonstop from Raleigh-Durham—every itinerary in business class from Raleigh to Bilbao routes through a European gateway, most sensibly Madrid via Iberia or London via British Airways, with Lufthansa's Frankfurt or Munich connection as a solid third option. Iberia makes the most operational sense given its Madrid hub and onward regional jets down to BIO, and its A330 business class, with lie-flat seats in a staggered configuration, is a comfortable long-haul product even if it lacks the polish of Middle Eastern or Asian carriers. British Airways routes through Heathrow, where Club Suite—with its sliding door and direct aisle access—has become one of the more consistently strong Atlantic products since its rollout across the 777 and A350 fleets; just budget extra connection time at LHR, particularly at Terminal 5, where transfers can eat 90 minutes even in good weather.
Having flown this corridor repeatedly, I'd steer experienced travelers toward the Iberia routing whenever schedules align, since Madrid-Barajas offers a shorter connection to Bilbao than the alternative of backtracking through London or Frankfurt, and Iberia's business lounge in Madrid's Terminal 4S is genuinely restful after a red-eye. Lufthansa is worth considering specifically when their Allegris cabin is confirmed on the transatlantic leg, though as of now that product isn't guaranteed on all Raleigh-compatible connections, so it pays to check aircraft type before booking. My insider tip: fly into Bilbao on a Friday rather than Saturday, since weekend leisure demand pushes Iberia's connecting fares up noticeably, and midweek arrivals give you first crack at the quieter, more atmospheric moments in the old town before tour groups descend.
Seasonality matters enormously on this route. January through March and again in October and November consistently offer the softest business class fares, as Basque Country tourism dips outside the summer festival season and shoulder-season business travel to Madrid creates more competitive award and cash inventory. Given how often fares fluctuate on this three-airline, one-stop corridor, setting up automated fare alerts is the smartest way to catch a rare dip before it disappears.