Your basic problem is that cheap fares still usually need a Sat night stay at the end destination or at least for your flight to that final destination leaving on the Saturday. A 5pm departure from ATH limits your options massively. Are you actually in Athens or are there other departure points that could be considered a little earlier in the day?

Found one wild routing, going an absolute mile back on yourself, if you have a strong stomach - EgyptAir/Aegean for $261 in Y to Cairo, then CAI-DOH-DFW return on Qatar, for $1980. Total price is roughly your budget, and flight times actually work for you.

CAI-DOH-CAI is in F, 60 TPs per leg, and DOH-DFW-DOH is in J, 140 TPs per leg.