So let's see if we can think of this another way -- if a President hostile to annexation (though not necessarily to Texas as such) is elected in 1844, how does the Lone Star Republic fare for the next four years? On the one hand, as
@drewmc2001 noted, they can most likely fight off any other invasions that Mexico may (or more likely, won't) throw at them; on the other hand, as also noted, their finances are in a very bad way, and pretty much the only ones willing and able to loan them money, Britain, want to see at least some move toward abolishing slavery as a condition for said aid.
Now it's noted that Texan political leadership took issue with this, but UIAM, that does proffer the question -- aren't these the same leaders who are at the very least highly preferential to annexation, and arguably have been banking pretty much everything on it happening? If it looks like said gamble is not paying off, and won't for at
least another four years, does the Texas political establishment become more receptive to some kind of manumission? Or are they, despite everything, still determined to carry on as an independent slave republic?
Not particularly. As a self professed expert on Texas

, allow me to explain. Sam Houston, when he was president of Texas, worked tirelessly on annexation. Politically, he was a unionist. He was also more sympathetic to the finding equitable solutions with AmerIndians. He was not virulently pro-slavery, like the fire-eaters across the south. But he had a very negative view towards abolitionism, and saw that as a destabilizing influence on the US. If someone is really interested in doing a TL on this, I may dig out some of the writings I found online between San Houston, Ashbel Smith, Anson Jones and Charles Elliott that I used when researching my own TL, that shows Houston's thinking on that issue. Hell, even Ashbel Smith, a Connecticut transplant, was proslavery according to his own correspondence.
But the problem is that Sam Houston's opponents were not abolitionists, not in Texas. His chief political rival was Mirabeau Lamar. Lamar was rabidly independence minded and was pretty much the 19th century equivalent of a white nationalist. My point is that without some pretty serious handwavium, it's hard to get to a Texas that would kowtow to Britain on the issue of slavery, unfortunately. Sure, you could reasonably craft a story that is internally consistent in which Texas agreed to limit the importation of slaves, from Africa. As early as the 1830s and 40s, it might even be enforceable. But any agreement would either exclude slaves from the United States being covered by the treaty.
@
RogueTraderEnthusiast, it's hard to overstate the case for cotton production between 1836 and 1860 in east Texas. Keep in mind, by 1860 in the US, chattel slavery counted for around $3 Billion in capital in the South, in 1860s currency. During the 20-30 years prior to the ACW, there was, I think, a correlation between the price of a slave and the global price of cotton. My point, is that these are the headwinds you're facing in thinking British loans would force a bunch of transplanted southerners into forgoing slavery during it's most lucrative time period.
Let's shift and talk about how to overcome these structural headwinds. It is, I think, about the POD. Expanded immigration from Europe helps. More Germans, more Scots, more French, more Scandinavian immigration during the years of the republic are needed to get Texans to think differently about Southern, racial slavery. In that environment, British loans
could have more of an impact, because there are more people who are not emotionally invested in slavery. Honestly, I'm not sure how to get that immigration up. The political uncertainty, Mexico's bellicose attitude, the panic of 1837 and other factors did nothing to make Texas a particularly inviting place to come. What Texas had was cheap land, and that's ultimately what drew folks to Texas during that period.