Almost a match: "The Censors" by Luisa Valenzuela
I was able to find a full-text for this story here (though I did not check for other translations).
Parts of the story that match provided details
It tells the story of someone who made their way up the ranks of the dystopia's mass surveillance agency screening mail.
That was the plan when Juan, like so many others, applied to be a censor. Not because of conviction like a few others or because he needed work like still others, no. He applied simply in order to try to intercept his own letter, not at all an original idea, but a comforting one.
This also establishes that the protagonist first decides to join the mail-screening organization in order to approve some important piece of mail.
Near the bottom of the ranks they just screen for toxins and poisons but once the mail clears one rung it gets passed up to the next where it is given increased scrutiny
... Section J where they unfold the letters with infinite care to see if they
contain poisonous powder ...
... he was assigned to Section K where the envelopes are opened with painstaking care to see if they contain some explosive.
... he rose rapidly until reaching E, where the work became more interesting, for there begins the reading and analysis of the letters. In that Section he could even cherish hopes of coming across his own missive written to Mariana which, judging by the time elapsed, should have reached this level after a very long procession through the other departments.
I interpret the last quote as implying that screening for hazards comes before analysis, but I did not notice a defined ladder of steps in the story.
I believe they are caught and executed.
The story ends with the protagonist's death
And just as naturally he couldn't prevent them from executing him at dawn ...
Parts of the story that do not match provided details
In the end, the protagonist makes it to the top of the chain of command so they can approve some important piece of mail.
The protagonist is never explicitly noted as being at the top of the chain of command, but is simply stated to be the most productive censor in the Censorship Bureau.
His Basket of Condemned Letters soon became the best nourished but also the most subtle in the whole Censorship Bureau.
After they approve it
In the end, the protagonist does not approve his letter.
He was at the point of feeling proud of himself, he was at the point of knowing that he had finally found his true path, when his own letter to Mariana reached his hands. Naturally he condemned it without remorse. And just as naturally he couldn't prevent them from executing him at dawn, one more victim of his devotion to work.
High School 2011-2015
(I am not sure if this section would be better as a comment, feel free to remove it.)
My confidence in this match is mostly based on remembering that I read this story in a textbook between 2006 and 2009 (middle school or high school).