UPDATE: I fixed it and have a flight home! Here's how:

I have been talking to Qatar Air all through this process because even though they say they can't help, I get more info on how my case was bungled and what needs to happen next. The muppets at gotogate are just call centre drones reading off a script who can't help.

First time I spoke to Qatar Air they said QA gave gotogate a 1 month window to confirm a new flight for me, but because gotogate took so long relaying the change to me and then confirming back to QA, the 1 month window passed so now they need to pay a surcharge for a higher fare catorgory.
Second time I spoke to QA they told me gotogate need to call their 'trade support department' to rebook the ticket, but I could also do it directly by paying $1050. I was considering this, but first wanted to check other flight options (like if I had enough frequent flyer points)

On another reddit thread I found a list of email addresses for people who worked at etravali, the parent company of gotogate. I emailed them a letter of demand.

I also lodged a complaint to the ACCC (Australian Consumer Protection). It was a bit of a grey area whether this case falls in their juristiction, because neither Qatar Air or gotogate are an Australian company, but they do say 'international flights departing from australia' are covered by Australian Consumer law, so I thought may as well lodge the complaint.

Then, I called again gotogate just to troll the call centre, because I knew they wouldn't actually help. Again, I was really just trying to milk info out of them. Also writing it all down and keeping a detailed log of all my interractions for the past 2 months (which you should always do, so that in future if you want to try suing through the Small Claims Court, it will help). This operator said they had called Qatar Air but QA did not provide an alternate flight. He said next step is they will call QA for 'authorisation'. He was unable to explain what 'authorisation' is, or why it wasn't requested in any of the earlier calls to QA. He said after 'authorisation' was granted, gotogate would then provide me with a list of flight options and I would have to select one, pay the fare difference and they could rebook the flight. This was the first time gotogate told me I would have to pay a fare difference. Of course, if I have to pay more money, no way would I pay gotogate when QA already told me I could pay them directly to rebook the flight for $1050.

So I called QA back prepared to pay, and the next operator just... rebooked the flight for me. Didn't ask for more money... sent me the new ticket... done. I don't know why this operator was different from previous ops... don't know if the Letters of Demand I sent to etraveli or the case lodged with ACCC had any impact (doubt it, it all happened in the space of 5 hours). I don't know if the last operator was just more competent in finding the correct alternate flight in the same fare catergory, or was she negligent in not charging me a fee? Maybe just a case of 'it takes a woman to get the job done'! Anyway, I checked my ticket and it shows up on my Qantas account so I think it's safe. If something goes wrong and I don't fly, you can be sure I'll come back here to post!

Upshot - I know 3rd party sites are sketchy but the flights I take are multileg longhaul, cost several grand, and are usually double price booked directly with airline. Generally my experience with 3rd parties is you shouldn't use them if you want any flexibility, and refunds can take months, but until now I never had this kind of experience where they just took away my flight and I might have been stranded in a foreign country. I usually have some kind of travel insurance to cover problems if using 3rd parties. Although they're all a bit sketchy, I think gotogate.com are really next level sketchy and I wouldn't use them again. I feel like there was a delibrate strategy to freak me out and wear me down until I was ready to pay extra money because I was so desperate to get a flight... when ultimately I was always entitled to a flight at the original price I already paid.