You don't necessarily need to keep Americans out of the Oregon Territory, just make sure that they are happy enough living in a British-ruled territory. Keep in mind that the Colonial Office would need to provide the American settlers with guarantees to protect their economic well being, physical security and eventually grant them self-government, to keep them happy enough living under the Union Jack. Most settlers would be driven by economic self-interest, that is the availability of free land, and the ability to prospect for gold, and would probably care little if they were under British rule or American, as long as their rights and freedoms were protected.
It is often overlooked that throughout the 19th century, Americans were the second-largest immigrant group to British North America, often surpassing British immigration during certain decades. They constituted the majority of settlers in the Prairie Provinces and by 1914 the majority of the inhabitants in Alberta were American-born or had an American parent. In Saskatchewan they were a plurality and in British Columbia and Manitoba they were a large minority.
American Immigration to British North America
1781-1790 50,000 (Loyalists)
1791-1800 50,000
1801-1810 50,000
1811-1820 5,000 (2,500 black)
1821-1830 2,000
1831-1840 8,000 (5,000 black)
1841-1850 30,000 (5,000 black)
1851-1860 60,000 (20,000 black)
1861-1870 10,000
1871-1880 30,000
1881-1890 30,000
1891-1900 70,000
1901-1910 458,000
1911-1920 625,000
1921-1930 198,000
Most Americans who settled in Canada seemed happy enough with the protections granted to them by the government there (including British Colonial Governments) inasmuch as there was never a serious movement by these immigrants to join the United States. This was in contrast to Mexico where the government was a bit more dysfunctional. Likewise a great deal of British Subjects people moved from British North America and Britain (even the English speaking regions) to the United States and seemed content enough with their new homeland and its government, once again because they were provided with rule of law and stability.
Foreign-born immigrants formed a major part of the early settlers to California for instance, especially once the gold rushes began. Here again are groups that could settled a British West Coast and could probably care little if they are American citizens or British subjects as long as their economic well being could be assured.
Therefore, the British could fill the West Coast up with immigrants, much like the Americans did. There were little restrictions on immigration during this period, and I can't see Chinese, German or even Mexican immigrants caring much about what flag they're under. Even the Irish immigrated in large numbers to British Australia, New Zealand, Canada and England and Scotland during this period, so I presume they would not rebel unless their economic well-being was threatened.
Keep in mind that in 1850, one-fourth of California's population was foreign-born, by 1860 that number increased to 38.5% with China 34,935 being the largest source of immigrants, followed by 33,147 from Ireland, 20,919 from Germany, 17,262 from Great Britain and 9,150 from Mexico, and 5,437 from British North America.