Check out Air Canada with CO/UA miles.
Not sure if this will work from LAX, but I just used CO OnePass reward ticket to purchase Executive First from LGA-YZZ-PVG, then TPE-NRT (this leg on ANA), then NRT-YZZ-LGA. All for 125,000 miles.
At least these Air Canada routes have 2 cabin classes: Economy and Executive First (lie-flat beds). For me coming from NYC it requires the stop-over in Toronto, but I think it's worth it.
Might also check out Korean Air because I know some of their routes also have only 2 cabins, allowing for lie-flat beds for business-class mile spend.
Good luck.