A couple of points from me.
You're well under the maximum 35k mileage which is a good sign and will give you lots more options.
Codeshares on non-oneworld airlines are not permitted which rules out a BA code on EI.
As Mumanddad suggested, try using the oneworld timetable on the oneworld website. Avoiding LHR would be a positive although I believe the APD only applies if you stopover rather than transit. That said, one option out of DUB may be with AY via HEL to somewhere in Norther America. There's also BA1 from SNN-JFK (I believe BA has uplift rights in SNN). On the way home, given you've got so much mileage up your sleeve, don't forget CX via HKG and JL via TYO, both of which can then connect in the other (ie. TYO or HKG) as well as other one-stop destinations to Australia including KUL, SIN, MNL and CGK.