Read the post regarding what the Republic of Texas required.
The United States and The Republic of Texas were negotiating as equals.
As sovereign nations they should have. But once Texas agreed to disband and join the U.S. and become a state then they were no longer equals. They were the Federal government and just another state.
The fact is, the Treaty you cited never said a word about secession and that treaty was never ratified by the Senate (only 16 Senators voted for it, so it wasn't even a sectional issue). It never had force of law. The treaty failed, BTW, because it required the United States to give Texas $10 million dollars to pay off their debt.
Texas was annexed the following year as a result of a simple Joint Resolution of Congress that spelled out the conditions Texas had to meet. Texas then proceded to meet all of those conditions Congress spelled out and was admitted as if it were already a territory. Those conditions BTW, required Texas to pay off all the debt they had accumulated to that point.
Four years after Texas was admitted, they finally got the US taxpayer to pay off their debts -- $10 million -- as part of the Compromise of 1850.
Aside from the provision that it could divide into as many as four addition states, with the approval of Congress, there was no rights granted Texas that were not granted other states on entering the Union.