Overall, I see Badgey as accidentally being the most sane one there and being a story of how sanity fares in an insane universe.
Here's my lore for Badgey and the ISS Cerritos (based on the IDW Mirror Universe and assuming that the mirror universe in Prodigy is the same timeline as IDW's mirror universe): I kind of imagine the story as the inverse, where the brutal Rutherford sends his Orion slave girl~1~ to test the simulation, which is a low gravity battle situation. Combined with Tendi's talent, the brutality and immorality scares Badgey enough that when the safeties are off, he gives his father a choice: help him take down the empire or face death at his hands. Rutherford immediately goes in for the kill, and the chase begins.
In No Small Parts, Rutherford seemingly convinces Badgey that the Pakleds are the greater evil, but tries to destroy the Cerritos with the Pakled ship, leaving his father to watch. Rutherford tries to detonate the warp core, but Shaxs, really wanting to do it, throws Rutherford into space (where he is beamed away) and detonates the warp core.
Finally, in A Few Badgeys More, Rutherford starts by trying to appeal to Badgey's hatred, which splits off into Mad-gey. He then appeals to the personal benefits Badgey will get, who splits off into profit-ey and is killed by Badgey. Finally, Badgey ascends and has the painful epiphany of how little good there is in the universe. He decides he will destroy the universe and recreate one prime among all the others, but notes the end might not be immediately apparent due to time crap.
1: On another note, my story for mirror D'Vana Tendi is as follows.
The Orions had a strong democratic socialist tradition for a long time until the Terran Empire conquered them. The Alliance later freed them from Terran rule, but did not allow the Orions their governmental tradition. Thus, the Tendi family became the leader of a secret independence movement to return Orion to its former ideals. This resistance continued even after Orion fell back under Terran rule in 2379 as part of the Terran fleet resurgence of the past few years.
With the Tendi family, D'Vana was known as Liberator of the Winter Constellations and was supposed to bring the revolution to a new age. However, she was known to be selfish and impulsive.
This came to a head when the Orion Resistance was going to attempt a major operation in 2380. Feeling it was hopeless, D'Vana made a deal with a Terran agent to sell out her rebellion in return for riches.
Like a true Terran, though, the agent lied, and the ISS Cerritos, assigned there on second plunder, took her and much of the rebellion as slaves... dismally easily.
Her sister, D'Erika, however, escaped and pledged not just to continue the revolutionary cause as the new Liberator of the Winter Constellations, but to get revenge on her double-crossing sister.
Meanwhile on the Cerritos, D'Vana was distributed to Billups. However, Billups preferred to work on improving the destructiveness of the phasers or watch his underlings squeal in the agony booth, so he decided to let Rutherford do whatever the heck he wanted with her. Thus, technically, she is not Rutherford's slave, but Billups, but ends up being forced to spend most of her time on the ship with Rutherford.
You’ve got quite a bit - it’s in TNG S4, so you’ve still got all of TAS (if you choose. If you’re thinking about skipping, at least watch Yesteryear), the first few movies, and the first three seasons and six episodes of TNG to go before you hit this glorious staple who will last through DS9.