TG is probably going to be your best option, at least as far as bangkok. From there you could pick up something like TK or any number of other carriers. TG only releases two seats at a time until a pre-determined limit is reached. That can be as many as eight seats per flight. Usually as soon as you book two another two will show, and so on. If the initial allocation has been exhausted TG often releases inventory close-in... tomorrow's SYD-BKK is a good example... it even has F available at short notice.
Once you are all in bangkok you might need to split the party - say two seats on Turkish and another on Austrian or Lufthansa (etc).
To get to Europe try TG the whole way, if not, try individual sectors and then the screenshot method. Use UA's booking tool to find availability, then recreate sector-by-sector on LM, screen-shotting what you need to.