BA has ultra high fuel taxes.
IB has fuel taxes, but much lower than BA.
Your most convenient trip is Iberia LAX-MAD and onward to Bilbao.
As you don't want that, use AA to fly to MIA or other airport on the east coast to connect with AA, IB, whatever. Use Aer Lingus with BA Avios. Etc.
You may need to travel two and two on separate itineraries; your chances of finding two award seats are ever so greater than finding four on the same flights..
Don't worry about going via Dublin etc. unless you need to for seat availability - you won't save airline imposed fuel taxes that way. You only pay United Kingdom Air Passenger Duty if you originate a flight in the U.K., unless you're connecting there under 24:00.