To avoid paying the extra segment why don't you fly BOM-BKK direct on CX? This is operated by the 744. Flight # is CX750 and operates on Mondays, Thursdays and Fridays.

Otherwise, skip one of your HKG transits (?) and fly BKK-NRT nonstop on JL. JL has three daily flights on the route, on a combination of 744 and 777. May be JALways operated, though, and I don't know how the first cabin looks like on these services.

You would lose some miles by doing so, but would not need to buy an additional segment in Asia and you would free up a sector in North America to fly LAX-DFW-YVR.