BA does not charge these egregious fees for redemptions on airberlin and Aer Lingus. So search for award availability on these two airlines.

Transferring avios to Iberia and booking those awards via them is also a good move. Iberia does charge "fuel" surcharges, but they are not outrageously expensive (expect something like $50 for a TATL).
However,you need to plan ahead, as there are some restrictions on transferring Avios between BAEC and IB+ accounts - the IB+ account must have been open three months and it must be active, that means it cannot be empty before you can move avios over from BA. Transferring Amex points into it prior to moving your BA avios over will satisfy that requirement.

As far as breaking up the flight goes - you will have to anyway (depending on your home airport). Airberlin and Iberia only serves a few cities in the US (LAX, ORD, JFK, MIA, plus some seasonal services in the summer, e.g.BOS). But you should be able to tack on an AA feeder flight to your award for no additional avios (if distance band remains the same).

Also, Bilbao has no direct transatlantic service, so you will have to connect in Madrid, Barcelona, Berlin or Düsseldorf as well.