Although I agree that it is likely the British lose the Pacific Northwest if they lose Canada, I don't entirely agree that it is absolute.
Nothing is ever absolute, but there is reasonable certainety.
If the ARW peace ends with Rupert's Land remaining in British hands (and more importantly, exclusive to the Hudson Bay Company),then you still have the HBC spearheading much of the exploration of the Oregon Country, which gives them a claim to the territory based on discovery.
Well, if Canada joins the ARW from the start, and we can reasonably assume that the British reap no greater military and diplomatic success than IOTL, America ends up the sole controller of the Great Lakes and would be greately interested into getting free settlement access to Rupert's Land. Therefore, I see only two likely outcomes: a) Britain and America agree by treaty to some form of joint ownership (like OTL Oregon) over Rupert's Land, or at the very least free access to American settlers and partially free access to US traders, even if the HBC may keep some trade privilege. In due time, American settlers swamp the area, and Britain agrees to sell RL b) Britain tries to keep sole ownership of RL, pressure by American settlers and traders grows to gain free access to the area, UK and USA fight TTL War of 1812 over this as well the other casi belli (impressment, blockade, etc.). War is fought over RL instead of Canada, only America shall be stronger and have a much better strategic position, whereas the British one shall be much worse. It is reasonable to expect that America reaps enough success that Britain agrees to sell or cede RL at once, or sees the futility of trying to keep it against American hostility and does so soon afterwards.
After all, when fur trade declines, the area becomes much less profitable for Britain, the mineral resources won't become known until much later, whereas America has strong interest in the area for agricultural settlement. And without Canada, the possibility of creating viable British colonies in the region becomes much worse.
Then, as the sea otter trade with China increases, that will give them the economic incentive to keep in that area, at least for a critical few decades.
That economic incentive seems rather marginal, and no worth risking the trouble of another war with a stronger America. When the fur trade declines, and without a Canada to latch it to, the value of Rupert's Land, Northwestern Territory, and Oregon-Columbia for the British Empire seems more and more marginal, esp. if the price is the hostility of America.
As to populating, if the British can settle Australia and New Zealand, they can send settlers anywhere!
But the vast majority of settlers for western Canada came from the rest of the Dominion. Britain actually was rather inefficient as it concerned populating its own White Dominions, and it does not seem credible that in addition to scrounging up the settlers for Australia, New Zealand, South Africa, and Kenya-Uganda, they can do so for populating western Canada from scratch, esp. in the face of the ever-growing pressure of plentiful American settlers to do the same.
On the other, other hand, they are at a disadvantage for competing for settlement of the region. If nothing else changes except they lose Canada at the end of the ARW and retain Rupert's Land, then they are likely to be outbred in the area. that means the Americans will take more of the Southern part of the Oregon Country than OTL, and maybe push British Columbia up above the 50th Parallel.
If America holds Canada from the start, it means they shall have sole control of the Great Lakes. They are not going to accept permanent British control and a barrier to their settlers as it concerns the valuable lands of Rupert's Land, Oregon, and Columbia, at least the agriculturally-valuable area between the 50° and 55° parallel.