Hey ya'll - wasn't sure whether to post this here or on the Finnair Plus forum. Just wanted to report my struggles going through this.
I managed to make this mixed AKL-(QR)->DOH-(AY)->CPH booking online just fine for AY's "new" seat. It was also a clearly labeled QR codeshare on the booking page, not sure if that made a difference.

Turns out I wasn't able to choose any seats for the AY flight. Neither QR nor AY's websites and apps wouldn't let me choose (AY website/app wouldn't even give me an option to pick seats).
I called AY and they said that they aren't able to pick seats for QR bookings until 48 hours before departure.
At T-48hr, I checked in through QR, and got a middle seat for AY.
I tried calling AY and they said that they can't do anything anymore since QR already _assigned_ a seat, I had to call QR to get it solved or handle it at check in.
Called QR and they also couldn't do anything.
Then I cancelled my check in, tried called AY... And they couldn't pick a seat either.
Note that when I cancelled my check in, my QR seat was unassigned and I lost it - luckily there was still a window seat available by the time I checked in.
When I showed up at AKL, the agent also said they weren't able to change my DOH-CPH seat... I'm in AKL atm so I guess I'll try again at the transfer desk in DOH, but for anyone doing the same, I guess you have to call AY _before_ checking in with QR.

Note that when you call AY, some automated voice message says that you'll be charged 1.7 euro per minute. I called using my google voice and never got this charge.