I can't ever recall this happening at check-in for BR at LAX and I fly them 4-6 times a year LAX-BKK-LAX. I have, though, called the LA office the day of departure to see if any op-ups or paid upgrades are available and for how much. If it's worth it to me, I tell them to hold the seat for me and I'll pay at check-in. Like I've said, never seen it done in the manner you described.