I like that idea too, but it would be more believable if the Thule had some experience with dealing with foreign powers like that. Based on their history so far, it seems the most likely reaction the Thule will have is to attack the invaders, and that strategy can only work for so long. Another possibility is trade-mediated infatuation with Europe and its culture, which is also a long-term bad idea for Thule civilization.
Would be be possible to retcon in some more nuanced politicking between Thule and neighboring peoples? Or between one Thule "state" and another?
I think that I've been making the point that Hudson Bay is a fairly nuanced place.
Let's take the Caribou Herding subculture. Here's the thing in the Thule realms, there's a perpetual low level struggle over land use and maximizing incompatible land uses. So caribou herders and farmers both seek the same productive land.
Obviously, they can't occupy the same land. That's where displacement wars come in, as each side pursues its comparative advantages, for control of land and land use. That's a simplification, but its along the lines.
Caribou herders are highly mobile, often across large distances, they can produce and sustain themselves on a lot lower quality of landscape productivity, and local adverse conditions don't really threaten them because they leave. They have lower densities, but they're able to access specialized resources across vast territories, and they're also able to use their mobility to concentrate large numbers for short periods.
On the other side, are farmers, they require a better quality of landscape but they're able to marshall extremely high productivity, to sustain dramatically larger populations. They're tied to lands however, are much less mobile and flexible, and local adversities can be catastrophic.
So anyway - that leads to displacement wars - cattlemen versus farmers.
But it's not all wars and bloodsheds. Generally, after a while, things settle out. Where one side has the clear advantage in one way or another, it wins, takes the land, and the other side goes elsewhere or dies. Hmmm. that's pretty morbid.
But things stabilize. Usually what happens is that farmers take and make use of the best land, and produce restricted areas of population density. The herders take the outskirts or outlying regions of either low yield land and isolated high yield land which is too remote to be defended.
Relegated to secondary lands and patches of good pasture, the herders have to move their herds around a lot to maintain. This is pretty standard practice.
What this means, however, is that the herders in their territories will be moving across and along the landscapes of several farming communities. So a kind of peace settles in. The herders who are able to get along with the farmers do better, they're able to trade, the farmers are less likely to shoot arrows at them, will 'lend them pasturage' and will support them in conflicts with rival herders.
Eventually, what evolves are proprietary 'herding routes' or migration routes, held by families or clans or tribes of herdsmen, which include not just the lands but relationships with a string of communities.
Now the tenor of these relationships can vary - herdsmen can be anything from no-account gypsies begging and scratching along the edges of the powerful farmers, or partners and travellers, or fearsome extortionists or tribute demanding lords.
Now, there's a couple of significant things to pay attention. One is that the herders are basically moving around a lot of meat on the hoof. Which means that if you're a farmer and you've just had a disaster, and you're on good terms with these guys... you'll be glad to see them come by.
But it also means that herders are travelling large distances with beasts of burden, and inevitably have a bit of surplus carrying or load bearing capacity. Basically, there's very little economic cost to them, in terms of their subsistence economics, to move a few tons a couple of hundred miles.... they already have the caribou, they can carry a load, and they're not carrying anything, so why not put on a pack or two...
This ability to move relatively larger or expanding quantities of items over long distances, is significant because it enhances the subsistence economy of the herders.
Instead of trade or exchange being exclusively local between neighbors, and trade movement being like a game of telephone - with goods moving through an infinitely long passage of hands, you have a class of people who are cutting out the chain of middlemen and can move goods a large distance between the producer and the consumer. More goods move, larger quantities of goods move, and they move faster. By being able to move goods between remoter distances, the 'value' of the goods increases and the herders harvest that 'value'.
A migration route evolves into a trade network, and if its valuable enough to the recipient farmers, they'll actually consent to expansion, surrendering or bargaining pastures, inviting the traders further, etc.
The Ellesmere Trading Network was the first great treading network among the Thule, and centuries later, it's still the biggest one, moving wool from Greenland, iron from Cape York, Bronze from Coppermine and Ivory from Alaska.
But in the Hudson Bay area, a number of regional networks have emerged, collectively carrying a more intense volume of trade, and working deals out among each other for control of territory, or exchange of products. Some of these are quite ambitious...
So, as far as trading and exchange goes, the Hudson Bay Thule are not a completely naive population, but contain constituencies which can see and possibly seize the opportunities that Europeans present.
This isn't even without precedent OTL. One of the reasons that Europeans were able to build their fur trading networks so rapidly, is that these networks already existed. The Europeans were actually repurposing local trading networks, which had exchanged things like flint and obsidian, shells, beads, copper artifacts, tobacco etc. What the Euros did was tap into the far end of the networks, and start pumping in hyper-valuable goods and increasing the volume dramatically.
The Hudson Bay trading networks are not as sophisticated as the Europeans, they don't even have the concept of money per se... However, we can take it for granted that they're much more sophisticated than the southern trade networks.