BA (conversion partner for Amex MR. Can buy AA MilesAAver flights), Delta, and Southwest have no close-in booking fees.

Please do your homework first though Do the below:

Search Delta, United, AA, US to see if any have saver availability. For Delta, if not the low tier, see if there are seats at the medium tier too. Amex MR's convert to it too. Compare to Southwest miles. Compare to cash (Amex MRs, along with cash itself, can also buy cash flights)

Is splitting up an option? Or buying 1 award seat and 1 revenue seat on the same plane?

Remember, when searching on Southwest, they fly out of Chicago Midway, not ORD. For LA, you can also consider searching for flights out of Burbank Bob Hope (BUR), Ontario (ONT), Long Beach (LGB), and John Wayne (SNA).