So, I was reading a little bit more into this, rereading Caton’s La Otra Historia de México (highly recommended) and I’m realizing that Iturbide committed some easily avoidable mistakes that would have otherwise kept him on the throne.
At the start of his Emperorship, Iturbide was very popular with the people, including most of the elites, who saw him as the best way to maintain the status quo while gaining independence. Even before Congress crowned him as emperor, Iturbide already held near-absolute power as President of the Provisional Junta.
From what I’ve read, Congress did not oppose Iturbide, because he was crowned Emperor; Iturbide was after all crowned by Congress, and Congress was initially formed by the request of Iturbide and the Provisional Junta; in a sense, Congress was returning the favor. Likewise the army – at least initially – did not specifically call for Iturbide’s abdication. Santa Anna’s initial Plan de Veracruz called for the restitution of Congress and the drafting of a constitution, not the removal of the Emperor. It wasn't until the opposition against Iturbide coalesced in the Plan de Casa Mata, that Santa Anna and company began calling for his abdication.
However, everything began to unravel because the Emperor and Congress could not agree on how to share power when they first sat down to draft a constitution. Congress saw itself as sovereign and the “keeper” of the legislative, executive, and judicial functions – separation of powers be damned – and the Emperor mostly as a figurehead, who had been crowned by Congress. Meanwhile, Iturbide wished for Emperor and the National Junta (think of it as his Cabinet) to retain executive power and for Congress to be a tad more representational (i.e. elected rather than appointed). It is easy to see how the drafting of the Constitution resulted in a stalemate. It was during this stalemate that Iturbide made several mistakes that could have easily been avoided.
Firstly, he promoted too many of his friends within the army too fast. This angered the established command and created the top-heavy officer corps that crippled the army for years to come. Iturbide could have found other ways to reward his followers and friends – say giving them large estates in the northern frontier – over army posts and promotions.
Secondly, when Iturbide reached a deadlock with Congress and chose to dismiss it, the Emperor became increasingly autocratic. By censoring the press and locking up as many dissidents as he could possibly suspect, Iturbide turned most of the intellectual and legal body against him. Naturally, this helped add fuel to the fire of those who opposed him.
And thirdly, in a stupid attempt to have the people rally behind him, Iturbide abolished several taxes – a policy that quickly drained the coffers – only to backtrack shortly after and increase property taxes instead. This became a surefire way to turn the elites and landowning class against him. Subsequently, without a healthy treasury, Iturbide was unable to pay the bloated officer corps he created in the first place. Slowly but surely the army then began to turn against him as well.
These mistakes could have and should have been avoided if Iturbide and Congress had reached an agreement. And Congress had been allowed to legislate on matters of taxation, freedom of the press, expansion of the army, and payment of veterans and officers.
Nevertheless, had things progressed as in OTL and Iturbide was still willing to go down the autocratic path, dismiss Congress, and risk inciting an upstart like Santa Anna to rebel, he might have still retained power by keeping the rest of the army happy. Iturbide’s final mistake in OTL, and the nail on the coffin to his emperorship, was sending inexperienced and recently promoted officers, like General Echávarri, to lead the charge against the insurgents. When Santa Anna rebelled, the Imperial army had already defeated Generals Bravo and Guerrero and had Guadalupe Victoria was pretty much surrounded. However, Echávarri ended up turning on the Empire and joining Santa Anna, which in turn freed Victoria and allowed Bravo and Guerrero to regroup. Within a few weeks, the tide had turned and the insurgents had the upper hand. Had Iturbide led the army himself it might have been enough to disincentivize further betrayals.
Remember that the goal of the rebellion, until the formation of the Plan de Casa Mata, was to simply reinstate Congress and let it decide on the style of government. Even after the Plan de Casa Mata formed, General Bravo swore to abide by the decision of Congress even if Congress decided to stay as a Constitutional Empire. However, by this point, Iturbide’s spiral into autocracy meant this was very unlikely. So even if the rebellion had been defeated and the army kept happy, Iturbide would be resented and it is unlikely the monarchy would have survived.
With this in mind, I think Iturbide’s root mistake was allowing a Congress to form before the Provisional Junta drafted a Constitution or at the very least an outline for one. All they had agreed on was to abide by the Three Guarantees and to not base the new government on Spain’s Cadiz Constitution. Iturbide and the Junta should have drafted a constitutional outline first and call for the election of a representational Congress while they still held near-absolute power and the backing of the army. Like in OTL, this Constitution would first extend the invitation to a member of a European royal house, and in the case that no one takes the offer have the Junta name a Mexican noble (i.e. Iturbide) as Emperor. It is very much like OTL but allows Iturbide to keep the upper hand from the get-go while establishing the much-needed separation of powers before anything else.
Anyways if there is interest, I might write a quick TL based on the outline above and my previous post. It won’t be a long TL, just a more detailed outline of my previous post, which continues in roughly that style till the present day (or as far as I get, since I have a habit of not finishing TLs due to RL getting in the way).