I don't think I'm proving your point at all. You brush off the rise of slavery in Texas from its inception as a republic to admission as if it is irrelevant, saying it wasn't 'entrenched,' but the slave population was rising BECAUSE the institution was widely supported! Your suggestion in the 'what-if' is that the Republic could have been induced to give up slavery by a U.S. that required it as precondition for admission. But the possibility that slavery would be ended was at LEAST a significant reason for Texas seceding from Mexico--after Santa Anna ripped up the Mexican federal constitution and extended the Mexican slavery ban to Texas, Texas erupted in rebellion. In fact, one of Texas' first actions was to ban free blacks from the Republic. In March of 1836 Texans put a slavery guarantee in their constitution, too. Hell, they even elected Lamar, who worked his fanny off for years to keep Texas independent and slaveholding. Lamar only ever came around to Union because he thought it would PROTECT slavery in Texas.
No, it does not at all seem likely to me that a country that had fought for its right to own slaves would suddenly acquiesce to give them up for the Union. Of course, we're arguing a what-if, and I don't want to belabor the point. We'll simply have to disagree. I don't mean to insult Texas by saying it, or demean the Texas Republic--I just figure different than you do about the purpose of Texan annexation to Texans.
I am not defending Texas or its use of slavery but you forget that the Republic of Texas was in a very weak position.
Assuming the U.S. president required a "no slavery requirement" for annexation, it had four choices: join the U.S. but give up slavery, rejoin Mexico (a very real possibility) but give up slavery, become a British possession (a real possibility), but give up slavery, or remain independent and vulnerable. Please note that President Sam Houston was very much a wild card on this as he was in 1861 when he came out against secession.