Unfortunately none of my sources I have at hand (I'm on the road at the moment) have that river's name. I can get it for you late in the week.
At that time, Gjoa Haven probably had open water for ~1 to 2 months at best, with access only from the east. Further west, I think the passage rarely opened up past Victoria Island at all.
Note: Most of these settlements only existed post-WW2, when the Canadian government rounded up the Inuit and moved them to those locations (and some others now abandoned). If there are historical maps from ~1930 for the Northwest Territories, which Nunavut was then part of, you should refer to them.
Hudson's Bay wasn't completely icelocked either, but note that I think the port of Churchill is something that only came around in the 1960s or so.