I know nothing about the story or anything else you've written, so this is a hot take.
You're writing in third person omniscient, past tense, which is a really common format, but requires some care to keep it feeling immediate. Particularly when you are using past tense to describe events that happened in the past. It can come across as telling, or info dumping, like a plot summary.
For example, the first paragraph:
It was still dark outside, but Beta had woken his two companions from their slumber already. They had decided to keep a lookout on the inner gatehouse from their fellow conspirer’s apartment, ostensibly to learn the sentries’ movements, but it also had the dual purpose of ensuring that no backstabbing or other unforeseen circumstances occurred during the night.
This is leaving a lot of opportunities for clues on the table and taking the reader's cognition out of the equation. Just something as simple as "had woken his two companions from their slumber." That happened in the past, and any information we readers could have gleaned is lost to us.
If you rephrased it to be current events, we'd learn so much more:
Beta merely whispered the word, 'wake,' and the others immediately rolled from their pads and crouched, tense and ready. << that tells me these are highly wired up professionals who are used to switching on in a heartbeat.
vs
Beta had to shake them several time to rouse his companions. 'I pushed them too hard,' he thought. << that tells me they are exhausted, and gives me anxiety like they might slip up.
vs
Beta woke his companions. Epsilon nodded and stretched. Nu had his dagger in hand. << tells me Epsilon trusts Beta, Nu doesn't.
By putting this in past tense but also describing past events without any character interactions, it feels really detached. This next sentence in particular is a doozy:
They had decided to keep a lookout on the inner gatehouse from their fellow conspirer’s apartment, ostensibly to learn the sentries’ movements, but it also had the dual purpose of ensuring that no backstabbing or other unforeseen circumstances occurred during the night.
There's a lot to unpack here. This sentence is telling us about a discussion between three people to make a plan, but also explaining post-mortem that there were hidden agendas, and furthermore hand waving to a tension so intense that Beta wondered if he'd even survive the night. That is a lot of subtext to summarize in one opaquely narrated past summary.
I want to see what they said, and observe the shifting eyes and vocal inflections that gave away the subtext. I want to see Beta and his fellow warriors nod to each other and sleep back to back, or tacitly agree to sleep in shifts with one hand on their daggers. Show me the tension that unfolded. It would be so much more immediate and engaging. It is eighteen paragraphs before we get to a single line of dialogue. Given the intense interpersonal dynamics here, that is far too long a wait IMO.
You could do all of that without adding to much to the bulk of the story, either:
Beta peered across the table at Epsilon and Nu. "We'll keep an eye on the gatehouse. Learn the sentry movements."
Nu said nothing. Merely narrowed his eyes.
"I'll take first shift," Beta said. As Epsilon passed, she raised an eyebrow, as if questioning the delay. Beta touched her arm and whispered: 'Sleep lightly."
The other side effect of this 3PO, detached POV is that the head-hopping is confusing. At times, Beta (or is it the narrator? impossible to determine) is describing what Nu likely is thinking, or should know, but it is not filtered through Nu, nor even Beta.
Take this, for example:
The old clerics of the phoenixes had been more than willing to divulge whatever they knew about the structure. Their simmering resentments made them eager to cause their overlords any discomfort they could, as long as their actions couldn’t be traced back to them.
Whose POV is this from? How does whoever's brain this is being filtered through know that the old clerics have simmering resentments? Did they voice such? Was it observing body language? How did whoever's POV this is figure out to approach the old clerics?
If you're going full-on 3PO, I suggest you put in way more observable detail and context cues for the reader to chew on. If you are taking a more limited POV (which in this case I suggest) then I would put the camera lens directly over Beta's shoulder the entire time and never break POV, and in fact lean hard into his interior thoughts and observations.
