Yes to both, but not the way people are saying it.
First of all, the idea of "flooding" areas with lands is so fallacious and downright silly in the 1830s that urgh... a great way to start making my blood boil

. America did not have this magical power to point and say "settle!" Not everywhere was Texas, and Texas largely grew out of a settler class given free
valuable farming land where everything's flat. The Oregon Country is dead-set between the Rockies, making it damned hard for anyone to actually get there. The Oregon Trail wasn't exactly an easy trek (just remember the game

) and when you finally got there; life was harsh and very wet.
Second, you have the fact that many Americans could give less of a shit who's flag they were claimed to be flying under out West. There was barely any authority as it was until the late 1800s there anyway, with only government forts (and later towns) resembling anything like the East. A fourth of my family comes from American stock in North Dakota, born and raised
for generations (since the War in fact), but came up to Canada in 1910 when we were offering a free 12 acres to anyone willing to give Saskatchewan a shot. Now we're unapologetically Canadian. Nova Scotia was built on 50,000 resentful settlers from Massacheusetts (who stayed and Canadianized nonetheless). Ontario only became full of loyalists after the war, before it was largely just a few farmers and frenchmen... I could go on, but I've made my point. Unless the local administration is corrupt, harassing settlers (which the British did not do but the Spanish did in Texas) and trying to enforce far-away laws, you're not going to get Texas-like settler conflict anywhere in the West. Now, don't get me wrong that I'm not saying settlement wouldn't
help it's just highly overrated in these scenarios as the "be all end all" to negotiations. There was plenty of American settlement north of the 49th IOTL, the region was almost entirely populated by American settlers and Hudson's Bay employees and yet it was split right down the middlde iotl: because these things are decided in board-rooms on the Atlantic coast, not on a demographic census of the region.
Finally, we come to where both PoDs can come in. In my honest opinion, if America invested more earlier settlement in the region and
made this known to the British (so they knew the region was virtually American settled), they are lead by a tough President and win quickly in Mexico and don't bring the British to brinkmanship but rather weather down their expectations, they might just attrition Britain into doing this. While keeping out of Mexico I don't think it's absolutely necessary they
don't involve themselves. Start a PoD with Polk being a bit more aggressive on the Oregon but not willing to fight over it. Negotiations are stalled while Texas ends up occupied, but enough interest in the region is kept up that settlement is helped along and the British are aware it's becoming more and more American. Then you have Texas fully annexed and the north looking for an appropriate counterbalance. Elect Zachary Taylor in 1848 and have him not die during the Presidency, so he can contain the South (if they rebel) and simulatenously not negotiate anything favorable to the British. In my opinion, whether nearer the end of his term or over the next decade (after a few aborted attempts at settlement themselves) the British might just take the attitude of "have it, whatever" and sign off the region to the U.S. This would have huge implications on future U.S. foreign policy though, as when they were tempered by the agreement iotl it was seen as a bit of a kicker to Manifest Destiny. The key isn't that the region is settled by Americans, but that Congress is encouraging it and the President is encouraging it; so it's a visible platform to the British whose leadership may simply give up caring about far off pacific domains rather than needlessly fight over it.
If you want the region to end up fully British, have 54'40 or Fight become a reality. It would go absolutely horribly for the United States in its infancy; Britain, at the height of her power, would blockade her ports and be able to occupy much of the territory without too much hastle. It would spout off further regional issues in the United States between the North and South (earlier Civil War?) and just be an entirely disastrous affair. Britain would probably end up with the whole Oregon Country and perhaps bits of Maine, etc. depending on if they bothered with land-occupation.