1) immigration would likely be massively affected, which in turn would affect the rate of western expansion.
2) taking west of the Mississippi is not an automatic thing.
3) the Texas scenario of moving in and then seceding to join British America is not an automatic thing.
4) You can't simply replace USA america with British America and expect everything else to go more or less the same.
1) Why? Canada received plenty of immigration per capita from the usual suspects in Europe, it was just lesser due to its proximity to better options (i.e. the USA) combined with a relatively small amount of pleasant-climate zones relative their neighbor; here, with a bigger BNA, I don't see that being an issue at all. And as mentioned before, having a bigger BNA means a closer option to mass-settle convicts to (say, north of the Great Lakes) than Australia....on the other side of the globe. In my estimation, that makes things at worst on par with OTL's USA, and in likelihood could lead to a bigger population.
2) If you look at the role large, traverse-able rivers play in the settlement and development of new lands, it really is. If you reach the Cis-Mississippi, you're within arm's reach of a major waterway (and control of same via New Orleans) facilitating further colonization (or even just commerce alone) that only a fool would hesitate taking, especially since the cost of doing so from Britain's POV is not appreciably greater than when they took Quebec in the Seven Years' War, and would certainly yield better returns on that investment in terms of arable land, access to minerals in the mountainous west, additional routes to connect cities and settlements together, etc.
3) Oh, the Brits would never do a thing like that, filibustering and/or co-opting other settled lands into their own. Just ask the Boers....oh, wait

. Again, nothing is set in stone, but it's also far from impossible as an occurrence.
4) Look, the details may differ, and it may not be specifically Texas that arises (and even if so, they may not join BNA at all) or any other major OTL territorial addition, but the way events unfolded in North America in the 19th century was due to a large number of reasons that the Brits would not be immune to. Manifest Destiny, as established in function if not title by the latter 18th century, would likely still continue (based on OTL trends in Australia and South Africa, why would Britain hesitate to take prime real estate and resources in North America), and there's still a ton of land even east of the Mississippi that would be attractive to immigrants to move to once towns inevitably sprout up (mild climate and relatively short travel time compared to the alternatives, plenty of work to be done by cheap labor by the locals, etc.). The British OTL track record towards native populations does not prove promising for the long-term well-being of Native Americans (Canada doesn't count, it's an outlier in that regard for various reasons), so I don't see that mattering much.