Only option for you is to issue PER-LHR on CX or BA and ask CX or BA to issue it. BA however may ask you to issue ith QF since their presence in Australia is limited to codeshare agreements with QF. So CX may be the only choice.

AFAIK, agents have to comply with issuing with the first outbound carrier. In real terms, one is not restricted to that, but in the TA world, one can only do as much as the airlines allow. Doesn't matter what WE say as to the rules etc, it's up to the issuing airline to interpret the rules they see fit as to maximise their bottom line.