Probably not, although in my opinion using the word “inevitable” when talking about history is, in itself, a very problematic thing to do. Historical change occurs through myriad contingencies and decisions and strategies and paths taken and not taken. Very little is inevitable, even if it appears that way in hindsight once it’s happened.
My point is simply that the whole battle over the west would have looked very different, and would possibly have had far more players than it did.
If neither of the new American nations had purchased Louisiana from France, for example, how would expansion have worked? Would both simply have begun to spread over French territory? Would one have sided with the French in order to gain an advantage over the other? If the French refused to sell, then the Americans (one or both nations) might have ended up with Louisiana anyway, but it would have been a longer and slower and more conflict-ridden process, which would, in turn, have shaped relationships in the continent as a whole.
What role would Britain have played, not just in the Oregon Territory, but in the more general disputes?
They didn’t actually try to keep them out. Empresarios like Austin were actually invited in, given land, and authorized to bring settlers, partly in an attempt to act as a sort of buffer against unchecked American immigration. The battle over Texas occurred not because Americans came in uninvited, but because the Americans who were invited in refused to comply with the conditions and expectations of the Mexican government. They refused to convert to Catholicism, they didn’t learn Spanish, many of them held Mexicans in contempt, and they brought slaves while, in some cases, presenting bogus indenture contracts in order to claim that they weren’t slaves at all.
Again, that’s entirely possible. I have not made the argument that what happened was the worst possible outcome. I’m not even arguing in terms of “worst” or “better” or “best” here, because i think that’s also a rather problematic way to try to understand and evaluate historical change. Worst for whom? Better for whom? Best for whom?