One option which is often missed is Iberia.

While they have only business class, no First, I rate their business class far higher than BA's Club World. And I love the food and wines.

They also have way, way lower fees and charges and taxes than BA if booked on the IB website. Same outrageous BA fees etc if booked on the BA site! (You can transfer Avios at no cost between BA and IB, and back again)

While IB have fewer North American gateways, Madrid is a good starting point for a European trip. Lots of low cost fares to/from there and Madrid a great European city.

Other options - no First, though - is Air Berlin, very very low taxes. Again, fewer US gateways. But you can use avios to book your AA flight from Texas too.

Finally, there is Aer Lingus for low fees and taxes to Dublin - another great European city. But again, no Texas airports, so an internal US connection. New Business seats very good. Way better than BA's CW.