Lots of people screaming "ASB". I think it's just very unlikely, but not altogether impossible. The trick would be to screw the USA and wank Texas at the same time.
I do think that the 1836 POD suggested by the OP makes things more difficult than they could otherwise be. Here's a scenario with a bit of an earlier POD (other than the Sobel one), since various other posters have already offered suggestions like that:
A dragged-out War of 1812 sees the North split off. Not just New England, but New York, Pennsylvania, Indiana Territory, Michigan Territory and Illinois Territory as well. Most settlers in those territories were from the North-East, which was fa more inclined to ally with Britain than the South was. The territoies were also threatened by Britain, so becoming part of a British-allied state (let's call it, predictably, the Federal States of America) would protect them. Assuming the war drags out, this scenario isn't unrealistic.
This leaves us a reduced USA that's basically everything south of the Mason-Dixon Line. Moving into the old North-West (now held by the damnyankees) isn't going to happen, at least not on the scale as in OTL. Southerners trekking west will be moving into... Texas. We can easily envision a Texan Revolution with more man-power being a bit more successful. This, in turn, attracts yet more Anglo settlers to more into Texas. Meanwhile, the USA and the FSA both want to move into the Louisiana Territory, but get in each other's way. Border skirmishes and the general lack of safety causes fewer settlers than OTL in the far North-West. The FSA settles the Old North-West more densely, earlier on, instead. The USA... moves more people into Texas.
Eventually, something like the Mexican-American War happens as in OTL. Both the FSA and the USA kind of want a foothold on the Pacific. Britain wants Oregon Country, too-- or at least all of the Seattle Bay area. Texas has its own ambitions. Stronger, more populated, it successfully participates in the war, its troops reaching California first. As the British-FSA alliance looks ready to best the USA, Texas cleverly opens negotiations with Britain. At the negotiating table, North America ends up looking rather like this:
(N.B. -- The map shows several border corrections I'd see Britain carry out after a dragged-out War of 1812.)
...so the USA is essentially screwed over by Texas, which wins big by doing so. The two nations are
not going to be friends anymore. Texas
is friends with both Britain and the FSA. taxas also has the best West Coast real estate, and is now a superpower in the making. Eventually, a world war erupts, with Texas and the USA on opposing sides. Texas utterly smashes the USA, conquers it, and begins the arduous process of trying to integrate the conquered territories...
Whether that process is ever truly successful is another question. A pleasant throught in this regard is that this super-Texas, absorbing so much non-slavery territory and being allied to two anti-slavery powers, would do away with the whole institution rather early on. If the rump-USA, consisting purely of slave states, presumably hold onto it, then the Texan "occupation" of the USA might just be a big campaign of liberation for the slaves.