Let's assume your default is via London, WT+ (T class) out and CW (I class) back, some mix of BA and AA, for EUR1800. Outward on 24th to meet the Sat night stay rule, and further assume you are happy with the limited flexibility.

Look at the routing, transfers and stopover rules for those fares. In both cases it's the mileage system, with max two transfers in Europe and two in the US. So any reasonable route on OW carriers would be possible. I'd focus on the return half where the CW fare will get you F sectors on AA domestics. Adding more sectors on a WT/WT+ fare will just add more surcharges and fees for few additional TPs.

A return half DFW-LAX-JFK-TXL-ATH looks interesting

I agree that leaving late on Saturday evening complicates things, unless you are happy to overnight in London.