The Bliss Jet looks to be a step too far for me - unless they have a guaranteed way to make good money with the aircraft on any "empty sectors". I didn't even know that LGA accommodated non-airline traffic. You wouldn't get very far across the Atlantic on anything but the smallest, shortest range private jet for $24k. But a first class ticket with a major carrier will compete well on price & schedule flexibility, even if you spend longer at the airports than using an FBO.
As for the Qantas flight, I'm sure I remember reading (on here?) a long time ago that they used to fly the odd charter to BKK and onwards to Australia for religious tour groups - possibly from MAN? Not sure if this could be a similar thing?