I've had a look at this, and do lots of work with TG, and it's a struggle to get them to offer MAN never mind GIB.
I can't see a fare beyond a full IATA fare, and would be amazed if one exists. I'm not surprised the agents wouldn't ticket it tbh.
What about LHR-GIB (BA) - Thailand (QR) - you can issue on one ticket with good connections on BA and good fares? Also you have access to QR's full Thailand network if your end point is not BKK (i.e. CNX/HKT/KBV) (although CNX & KBV on the return is a day flight and won't get you to GIB in one day as arrives in BKK too late).