Yes I have heard of mobile welders....
I was also going to suggest Adult education (not TAFE). I did a one semester course in Arc welding at Arthur Philiip in parramatta (my closest venue that did it at the time), but it is a regular night for a number of weeks - I'm sure these days they would also do MIG.

The other suggestion is that I did a one day course in Mig welding at the (then) CIG training centre years ago (which was in Regents Park from memory). They are now BOC, so ring them and ask if they still do it. I did both mild steel and aluminium. I happened to be the only one booked for that day, but they run classes for up to about 10 people I think. Back then ('92-94ish) it cost about $150 or so.
KB

A few years ago I did a panelbeating handskills course at tafe which included welding , it was one night a week and the entire class was car nuts who were restoring cars and wanting to do a lot of the work themselves . The course wasn't a trades qualification one but an interest course and most tafes will have something similar . The best bit was that it was a 2 year course , first year you did some of their projects and some of your own , the second year it was all your projects and had full use of their equipment and materials . Strange how most of us kept failing and redoing the second year as well ..... See what's available around your area from all options (Tafe , Adult Education etc ) , hopefully there'll be something handy .

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In my job at Rheem I used to organise welder training in MIG and stick.
Originally we used Lincoln Electric, then Cigweld.
Both companies have now got out of the training scene, in part due to insurance liability.
When Rheem sends people now they go to TAFE at Chullora, there is a 1 week course they have set up for us.
I'd ring TAFE at Chullora and see what they can offer you.

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DrMini- 1970 wasaMatic 1360, Mk1S crank, 86.6HP (ATW) =~125 @ crank, 45 Dellorto (38 chokes), RE282 sprint cam, 1.5 rockers, 11.0:1 C/R.