Well a cause of Texas's independence and the eventual Mexican War is that México outlawed slavery, whilst most of the Americans that settled there (becuase they were invited to by México City) were slaveowners. Butterfly away any chance of Americans settling in México (which could require a near-ASB change in México's political situation at that time), which would include the Texas annexation, the Mexican War and eventual Cession, and such a thing would be possible. However, being negros, they would occupy a low position in México's class system, so if ex-slaves settled in México they would really not like their position and would revolt.
Actually, the American settlers who were legally approved by Mexico City were the ones least supportive of secession from Mexico: until he got roughed up, Sam Houston was the foremost opponent of separation. By and whole, the settlers that entered Texas with mexican permission had the following characteristics: the were Catholic, swore to become solely Mexican citizens and let go of American citizenship, and promised to obey the Mexican laws. Even approaching the eve of the Texas Revolution, there was still an strong current for compromise; Santa Anna was just to stubborn/foolish/ignorant/corrupt (your choice) to do so.
You have to remember that slave-holder were never the same kind of force in Texas, even once in the US with the free movement of slavery into Texas, as they were even in the most marginal slave-states; Texas prairie isn't good for much slave-work, and ranching (the Mexican-era money-maker) is not a especially good slave buisness.
Instead, it was the hoardes of illegal settlers that went into Texas
without permission. The prblem with the Mexican Approach was serious: not only did you make a serious requirement of requiring people to change religion, but you also limit yourself to those who swear to uphold your law. Which is good... if they are the majority. But when it is well known that you can't guard the borders (as is impossible over the prairies) to keep the law-breakers out, the only people who won't come in are the law-abiding citizens, ie the ones you would actually want.
The popular support and base of Texas independence were the thousands of individuals who not only rule breakers/ignorers by nature, but also never stopped considering themselves American by nature. (Hence the complaint by Mexican officials trying to enforce their laws that every settler seemed to have a copy of the Constitution in their back pocket.)
Getting rid of the slaveholders would be the equivelant of cutting only a few heads from the mighty Hyrda. While richer than non-slave owners, they are a marginal fraction of the population base. And while slave-owner was near-synonymous with local leader in the Deep South, it not only gets less-so out west in the Plains, but Texas already had a number of leaders who were, surprise surprise, frontiersmen and not slave-owners.
Trying to ban all settlement as well is equally futile. More so, since if the US in this day and age can't secure the border to such a degree, how is poor Mexico supposed to do so back then? Even before the settlers came, traders who worked out the best methods for crossing the Texas expanse were making illegal runs into the cities of Texas, creating a cultural prejudice that Mexicans were lazy and corrupt as traders told the tales of the Siesta and how they could bribe the few poor local officials to ignore them as they came through.
If you want to keep Texas from seceeding, lower the immigration restrictions. Don't keep out the law-abiders because they won't convert to Catholicism, and don't refuse to compromise. Or better yet, don't let Santa Anna play tiny dictator and drive away those idealists at the fringes of Mexico.
But onto the original question, I would argue that the Underground Railroad would still exist, and Canada would still be a destination, for the twin virtues of (a) almost definitely not being (a) slave state(s) and for being such a far distance from the South. In fact, there might be even more escapees than IOTL; where as Canada gained a negligable dribble of OTL as most slaves stopped elsewhere in the US, an Underground Railroad that can freely move across the (nonexistent) border can have more agents and more connections in the settled areas up north from to which slaves could be shuttled. Bounty hunters could also go up easier as well, but it wasn't like none ever went across the border IOTL either.