Seems your TA perhaps didn't look too hard? The following schedule QF & BA all the way (one way) 86,000 points in Y, straight off the QF website (note a Saturday flight showed AY being the only flight available, but changing dates soon helped that):
Your Itinerary : 1 Adult
[TABLE="class: full"]
[TR]
[TH]
Flight 1[/TH]
[TH="colspan: 2"]From[/TH]
[TH="colspan: 2"]To[/TH]
[TH]Flight[/TH]
[TH]Departure
Terminal[/TH]
[TH]Travel Class[/TH]
[TH] Baggage[/TH]
[/TR]
[TR="class: tr_first"]
[TD="class: nowrap"] Tue 8 Oct 13[/TD]
[TD]
06:00[/TD]
[TD] Adelaide[/TD]
[TD]
08:25[/TD]
[TD] Sydney[/TD]
[TD] Qantas flight
QF730[/TD]
[TD="class: aligncenter"] 1[/TD]
[TD] Economy[/TD]
[TD="class: baggage"] Checked:
1 piece[/TD]
[/TR]
[TR]
[TH]
Flight 2[/TH]
[TH="colspan: 2"][/TH]
[TH="colspan: 2"][/TH]
[TH][/TH]
[TH][/TH]
[TH][/TH]
[TH][/TH]
[/TR]
[TR="class: tr_first"]
[TD="class: nowrap"] Tue 8 Oct 13[/TD]
[TD]
10:40[/TD]
[TD] Sydney[/TD]
[TD]
16:35[/TD]
[TD] Bangkok[/TD]
[TD] Qantas flight
QF23[/TD]
[TD="class: aligncenter"] 1[/TD]
[TD] Economy[/TD]
[TD="class: baggage"] Checked:
1 piece[/TD]
[/TR]
[TR]
[TH]
Flight 3[/TH]
[TH="colspan: 2"][/TH]
[TH="colspan: 2"][/TH]
[TH][/TH]
[TH][/TH]
[TH][/TH]
[TH][/TH]
[/TR]
[TR="class: tr_first"]
[TD="class: nowrap"] Mon 14 Oct 13[/TD]
[TD]
00:20[/TD]
[TD] Bangkok[/TD]
[TD]
06:25[/TD]
[TD] London (Heathrow)[/TD]
[TD] British Airways flight
BA10[/TD]
[TD="class: aligncenter"][/TD]
[TD] Economy[/TD]
[TD="class: baggage"] Checked:
1 piece[/TD]
[/TR]
[TR]
[TD="colspan: 8"]
- = Flight Operated by British Airways
[/TD]
[/TR]
[TR]
[TH]
Flight 4[/TH]
[TH="colspan: 2"][/TH]
[TH="colspan: 2"][/TH]
[TH][/TH]
[TH][/TH]
[TH][/TH]
[TH][/TH]
[/TR]
[TR="class: tr_first"]
[TD="class: nowrap"] Mon 14 Oct 13[/TD]
[TD]
20:30[/TD]
[TD] London (Heathrow)[/TD]
[TD]
23:55[/TD]
[TD] Budapest[/TD]
[TD] British Airways flight
BA870[/TD]
[TD="class: aligncenter"] 3[/TD]
[TD] Economy[/TD]
[TD="class: baggage"] Checked:
1 piece[/TD]
[/TR]
[TR]
[TD="colspan: 8"]
- = Flight Operated by British Airways
[/TD]
[/TR]
[/TABLE]