why do ex EU if cost don't matter?
EDIT: just realised you said it needs to be less than £3k. that probably does mean ex EU as I think most ex UK fares are in the £3/3.5k range.
from memory, you an build extra stops into the ex Norway fares for extra money so you may be able to stretch Norway-LHR-SIN-HKG-PER-SYD (or even more with some meandering around Asia or OZ e.g. SIN - BKK - HKG or HKG - PER BNE SYD)