Ah, no.
The full brunt of criticism and complaints should be levelled at BA.
QF have done little if not nothing wrong. Their website accurately represents what are the correct guidelines for access to the lounges. The agreements were made some time ago and both airlines, "on paper" and at a high level, are aware of the implications of it. It was the incompetence of a BA lounge staff member (or multiple, as it would seem) that resulted in the incorrect action. The core fault lies with BA, not QF. The only reasonable exceptions would be (a) capacity control, or (b) change in policy which excludes QF WPs from accessing the BA T5 GF. The former may be understandable and QF would have no control over this anyway. The latter is in direct contravention of the rules of the alliance, but again is not QF's fault because they have correctly said their members are entitled (as opposed to incorrectly saying that their members are entitled, in which case QF would have a problem).
Raising the complaint through QF will not work well anyway; it is inefficient because it relies on a secondary process between someone at QF and someone at BA to raise the issue. A more efficient method would be to raise the complaint with BA directly. It is probably worth raising the issue with QF if only for tracking purposes (i.e. if there was a pattern of many QF WPs being rejected from BA F lounges then it may be a serious problem), although BA should have a process / register of complaints that would be able to pick up similar patterns. Of course, if BA truly know what they are doing then complaints of this type (lounge access for oneworld Emeralds) should never be raised in the first place!
Now if in raising the issue with BA they tell the OP where to go (again, incorrect thing to say to a oneworld Emerald), then I might consider getting QF involved but it will still be an inefficient process. At this point I would also start questioning the competence of BA much more seriously more than why QF aren't doing their job throttling BA into honouring their terms of the alliance.
I did say too, not instead of. Am not by any means suggesting that you should not complain to BA but can't agree that Qantas have done nothing wrong. Under Australian law (presuming that where the OP is from) if you say you will do something under a contract there is a funny little idea that you actually have to do it, regardless of whether this is outsourced to a 3rd party. So I object to the we can't/won't do anything attitude.
As for efficiency well would have to seem there is no evidence in this thread to date that complaining to BA has been either efficient or effective. The OP did point out their entitlement and it was apparently rejected out of hand.