ASP doesn't operate all hours (as far as I can tell, why would it) and has no customs facilities (mostly a guess, can't be far off). I don't trust the Australian government (via Border Force) to permit a plane load of passengers off the aircraft onto Australian territory without formal processing, not for a medical emergency let alone this situation (pressing, but admittedly less urgent).
If the path is as you suggest, perhaps the only options are PHE, DRW or PER, though if the call were made above ASP, arguably the call to continue to BNE is technically plausible.