Contact the school and see if they would be willing to somehow work with you on the payment of a 1 month deposit (e.g., pay 2 weeks up front in advance of your trip and then pay remainder part way through). The school is giving you a rock bottom price even at 430 Euro for a month.
A bunk bed in a shared dorm room at a hostel would cost you about 20-25 Euro a night, and the majority of hostels won't let you book a bed longer than 1 week, maybe 2, tops. So you'd have to keep switching hostels every week or so. Maybe you could find a cheaper hostel way out in some suburban area, but then how are you going to get around because your public transit costs just went up.
I just don't see any one couch surfing host wanting someone staying with them free of charge for 5-6 weeks, so you'd have to continue to keep your fingers crossed another couch surfing spot became available somewhere else. And I just can't envision a lot of couch surfing hosts would be ok about your using their kitchen every day to cook your meals. It seems most hosts just want to welcome someone to flop on their sofa for 1-2 nights, tops (with the unspoken expectation of perhaps your buying them dinner or taking them out for drinks in exchange for the free lodging).
Why not just take training like this in Auckland or Christchurch if you think you can find cheaper or free lodging in your own backyard? Are you not able to get a refund at this stage if you cancel the training in Paris?
You could try to find something on Craigslist's room shares, or post what it is you are hoping to find with your limited funds and see if you somehow manage to get a response. They might have something in your price range, but I would guess it's going to be way outside of Paris central (and you'd need to check very carefully on who it is you might be dealing with since it's not any sort of arrangement that's been vetted through the school).