If you can get the roundtrip for < $700 per person, and don't need the flexibility to change or cancel that comes with some award tickets, then paying cash for this trip sounds like the way to go. Just make sure that you are comparing apples and oranges. Does the < $700 fare allow you to check a bag? Select a seat?

If you purchase cash tickets via the Amex travel portal, what value from your MR points would you get if you opted to pay with points?

Neither AA nor UA collects surcharges on their award tickets between the U.S. and Asia, and each offers LAX-TYO-LAX for 70K miles and minimal fees in Economy. But Amex MR points do not convert to either program; and even if they did, you'd still be getting less than 1.0 cent per point in redemption value, based on the current cash fare.