I'm looking at holding a portrait exhibition of approximately 340 prints (around A3 size)
That sounds like a really huge exhibition. I realize this is off-topic, but where are you going to exhibit 340 prints of A3 size? And who wants to view an exhibition of 340 portraits?
[A]fter realising the cost of this many prints with a professional printer I'm now looking at purchasing a printer and printing the series myself. Wanting to spend under $1K.
For the Pro-100, Red River Paper's test found ink use of 1.2ml per ft^2 (see
https://www.redrivercatalog.com/cost-of-inkjet-printing-canon-pro-100.html). So 340 prints of A3 size would need about 549ml of ink before any is used for self-cleanings. For the Pro-100, I'd assume at least 15% more for self-cleanings. Pro-100 cartridges are 13ml, so you'd need about 49 cartridges. On Amazon Australia the cartridges appear to run at least $20 AU each, so the ink would probably consume all of your budget before you spent anything on the printer or paper.
For the Pro-10, Red River Paper's test found ink plus chroma optimizer use of 1.9ml/ft^2 (see
https://www.redrivercatalog.com/cost-of-inkjet-printing-canon-pro-10.html). So 340 prints of A3 size would need about 869ml of ink before any is used for self-cleanings. For the Pro-10, I'd assume at least 30% more for self-cleanings. Pro-10 cartridges are 14ml, so you'd need about 81 cartridges. On Amazon Australia the cartridges appear to run at least $21.50 AU each (see
https://www.amazon.com.au/Canon-PGI-72-Ten-pack-Ink-Tanks/dp/B00AV8MUAY/), so the ink would very likely well exceed your budget before you spent anything on the printer or paper.
For something like your project, a printer like a Canon Pro-1000 (which comes with 880ml of ink and 80ml of chroma optimizer) might be more economical. But I still don't see it fitting your budget.
ideally on a matte to semi gloss paper (any paper recommendations also welcome)
Given your budget, if you get a Canon printer, I suspect Canon SG-201 Photo Paper Plus Semi-Gloss is probably the best bet for good quality (albeit RC-type) and reasonably-priced. But A3 or A3+ sheets will probably cost you $1 AU to $1.50 AU each, so that's another $340 AU to $510 AU, on top of the printer and ink.
I would appreciate any advice ....
I would consider scaling back from 340 portraits to maybe 34. If you really want or need 340, then I would look for a budget-friendly lab that prints with a web process (RA-4) and would print A3 or A3+ on a semigloss or luster-type paper. In the U.S. I think I could get the prints for about $5 or $6 each, especially in that quantity. The time and effort would be much less, and I suspect at the end of the day, the cost would also be less.
Good luck!