Here are the relevant articles of the treaty (and note that Mexican citizens became US citizens unless they chose otherwise and in which case they had to leave):
ARTICLE VIII
Mexicans now established in territories previously belonging to Mexico, and which remain for the future within the limits of the United States, as defined by the present treaty, shall be free to continue where they now reside, or to remove at any time to the Mexican Republic, retaining the property which they possess in the said territories, or disposing thereof, and removing the proceeds wherever they please, without their being subjected, on this account, to any contribution, tax, or charge whatever.
Those who shall prefer to remain in the said territories may either retain the title and rights of Mexican citizens, or acquire those of citizens of the United States. But they shall be under the obligation to make their election within one year from the date of the exchange of ratifications of this treaty; and those who shall remain in the said territories after the expiration of that year, without having declared their intention to retain the character of Mexicans, shall be considered to have elected to become citizens of the United States.
In the said territories, property of every kind, now belonging to Mexicans not established there, shall be inviolably respected. The present owners, the heirs of these, and all Mexicans who may hereafter acquire said property by contract, shall enjoy with respect to it guarantees equally ample as if the same belonged to citizens of the United States.
ARTICLE IX
The Mexicans who, in the territories aforesaid, shall not preserve the character of citizens of the Mexican Republic, conformably with what is stipulated in the preceding article, shall be incorporated into the Union of the United States. and be admitted at the proper time (to be judged of by the Congress of the United States) to the enjoyment of all the rights of citizens of the United States, according to the principles of the Constitution; and in the mean time, shall be maintained and protected in the free enjoyment of their liberty and property, and secured in the free exercise of their religion without; restriction.
Our United States federal government did not purchase land from Mexico. Mexico was defeated and there was no obligation to pay them anything. However, our federal government in the interest of adjusting the border for jurisdictional control and to indemnify against any future lawsuits and to facilitate the Mexican government exit from the territories agreed to pledge $15 million in installments over time. It was never about purchasing land.
It should also be noted that Mexicans before their revolution had a landed gentry who in turn owned 'Haciendas' that were larger than many US western states. The Treaty specifically points out that their land ownership will be respected. That means they continued to hold title until they sold.
The land is jointly administered by BLM and the state of Nevada with Nevada holding states rights protections. Nevada has as a choice to haul the BLM into state or federal court and sue for damages. Bundy is hence armed with plenty of legal options in which to pursue damages against the BLM.
There's a twist; there always is.
Nev. Rev. Stat. 568.010-568.210 recognize federal Taylor Grazing Act districts and allotments and set up a state, district, and county system to allocate the funds returned to the State of Nevada under the Taylor Grazing Act. The only administered right is the allocation of funds received from the federal government.
Nev. Rev. Stat. 568.225 is qualified "except as provided in the Taylor Grazing Act."
Multiple federal courts have ruled Nev. Rev. Stat. 568.230-290 violate the Supremacy Clause of the U.S. Constitution if applied to federal land.
Federal judges with lifetime appointments interpreting federal laws and the U.S. Constitution based on argument made by federal lawyers representing federal agencies with federal agendas. See Marbury v. Madison.