Help with denied ESTA - Previously issued DV LPR but did not activate it?

I'd primarily wanted to keep it on hand as an option if I did want to work elsewhere in the future or my life circumstances changed further.

How? Last I checked, US immigrant visas are only valid for 6 months, after which you have to apply anew (and with DV visa - you'll need to win the DV lottery anew).

Is there anything I can do outside of apply for a B2 or something similar? Wait times for the Sydney consulate are sitting at almost 3 months currently.

No, if ESTA is denied you'll need a visa.

Do I need to fill out this relinquishing of LPR status form I keep reading about in similar questions even though I never activated my visa?

From what you described, you haven't relinquished your LPR status because you never had it. All you had was a visa to enter the US as an LPR which you never used.

Presumably your immigrant visa (ie, your 'Diversity Lottery' visa) has expired, as they expire 6 months after being issued. You then applied for an ESTA which was denied.

It's impossible to say with certainty, however the visa is most likely the reason your new ESTA was denied. By applying for an immigrant visa you are deemed to have shown "immigrant intent", which basically means that you've shown that you have an interest in living in the US permanently.

Having shown immigrant intent (and then not actually immigrating to the US) does not preclude you from entering the US as a non-immigrant (ie, under the Visa Waiver Program/ESTA, or with a non-immigrant visa), however it does increase the level of scrutiny you will receive as there is an implied presumption that you are intending to move to the US permanently as you have already shown a desire to do that by apply for an immigrant visa. This is most likely why your ESTA application was denied.

Your only option at this stage is to apply for a non-immigrant visa (eg, a B1/B2 visa) which needs to be done in person at a US Consulate. You should be prepared to show as much proof as possible that you do NOT intend to attempt to reside in the US, and that you will return at the end of your trip, as a means of overcoming the "immigration intent" that you've already displayed. This proof could include proof of a job in your home country, proof of ownership of a house, etc.

Even once you've received your visa, you may also be subject to additional scrutiny when actually entering the US for the same reason. Although a return ticket is not required when entering the US on a B1/B2 visa, it would be recommended to have one as further proof that you intend to depart the US at the end of your trip.

There is no need for you to relinquish your Permanent Resident status, as you never actually had this status. In order to actually become a US Permanent Resident you would need to have entered the US using your immigrant visa before it expired. As you never did that, you never had Permanent Resident status.

Wait times at US Consulates in Australia are indeed high at the moment. Perth is generally the shortest wait, however is obviously a long way away from Sydney (and there may be quarantine requirements when coming from another state). Appointments do become available at all consulates at short notice due to cancelations, so it's worth checking frequently.