This did cross my mind, which is why I bought new belts. I inspected the old belts and the conditions weren't too bad, so I kept them in case I needed it. Glad I did.
The new AC belt was incorrect. To Vass' point, I needed one with an extra spline. I put the old belt back in and the sound is gone.
Pulleys aren't exactly mint so I think I will replace them soon too. The corrosion was quite nasty, so I will be putting fresh coolant in soon, but I think I'm going to disassemble the system again in a few weeks to check the AC and thermostat holes and see if they're clean.
I've had the car for a few months and it always had a whiny AC too. Looks like I will be replacing a bunch of spinny things in the near future.
Thanks to all of you who came to help! I've yet to do a proper test drive so it's not quite over, but idling is perfect. Will be back soon when something else inevitably breaks.