I highly doubt that the British could have taken all of Oregon territory. Following the Louisiana Purchase America was to big a player to ignore. There were also more Americans entering it as far as I know than British. Thats one of the reason the split occurred OTL.
It's worth pointing out that the Louisiana purchased wasn't clearly defined in the north and Spain disputed its extent also. Further treaties with Britain and Spain after 1812 were needed to set these. Have the US come off worse with Britain and you'll get a more southern boundary than OTL. It's worth noting that the settlers weren't exactly US loyalists/republicans, plenty became Canadian as they ended up in Ontario to British Columbia etc.
But still makes it extremely difficult for the entire territory to be British. And even a victorious war with the US over it would just push BC into joining a Confederation out if self defence.
So I agree that the Louisiana Purchase as it is reduces the likelihood of a British entire Oregon/Columbia.
Perhaps the best way might be to somehow get a US alliance with Britain to conquer it from Napoleonic France. That way the US never gets the northern portion bordering the Columbia/Oregon territories and the US expansion west is limited by wars with Spain with or against the UK and so is much slower. Without a race to the west with the US British America has no need to consolidate east to west.