USA to Warsaw, early July - 210K AA miles

I'd prefer to fly out of DFW as it's closest to where I live (and a large AA hub), but I can get anywhere in the US if needed. I'm trying to avoid massive BA surcharges, and it seems to be almost impossible.

I need to leave on 7/1 or 7/2 and be in Warsaw by mid-day on 7/3.

This is my first time using miles internationally, so I could use all the help I can get trying to navigate to not pay those ridiculous BA fuel surcharges.

I see availability for award travel on AirBerlin (A Oneworld partner) that would fit with what I need, but only with Avios through the BA booking tool. Is there a way I can book AirBerlin rewards flights using AA miles?

Thanks!