I used 160 K to fly AF Concorde on 5/21. I had called almost daily this month and there were no seats. Thanks to a FT posting last week, I snagged one! If you can work out the mileage issue, I would keep calling AA/ Qantas, etc to see if one opens up. As you can imagine, they are quickly snatched up.
It is TRULY a ONCE IN A LIFETIME experience!! Give it your best shot! What a great anniversary present!