I think I might end up in a similar situation as I'm looking to book an ex-CPT DONE5 in the next month.

If I can't book through AA (I'll give it a try at least), is CX the next best option to avoid high fuel surcharges, or should I just go with BA?

As to whether you can book a codeshare standalone. I know I was able to do this a few years back when AY code-shared on QF SYD-BKK as part of a Circle trip. However, as I'm looking to fly MAD-GIG for my transatlantic, even adding a AA codeshare CPT-LHR wouldn't fix the problem.