The ship’s last two numbers of it’s registry is 31, It's a blatant Easter egg. Also the fact that the Discovery hides in a secret asteroid base and the black badge personnel seems to lean to a Section 31 covert operation. The black badges could be Section 31 agents guarding the ship and it’s mission.
I have to add something else Lorca could be Garth of Izar. From the little we seen of Lorca it’s becoming clear that the man is unhinged and maybe laying the ground work for the starting of a psychosis like Garth, and it certainly fits the timeline; and Garth’s mental break down later on. The name Lorca translates into "Lord" in Czech and Bulgarian, i.e Lord Garth, a title he takes later after his severe break down of reality.
Garth has a lord complex and straight from Star Trek TOS: episode Whom Gods Destroy. In 2268, Garth managed to overpower the colony's keepers, proclaimed himself Lord Garth, master of the universe..." So there could be some loose tie in there. Perhaps Gabriel Lorca is a code name under Section 31. Garth and the name Lorca could be the equivalent to a Lord which we know from canon Garth proclaims himself as.
I think that whether or not the show outright comes out and reveals him to be Garth, I strongly believe this is the character they are modeling Lorca after. He already exhibits many of the traits of Captain Garth (he's an accomplished Captain, military tactician fighting a war, takes place around the same era as the TOS episode introducing Garth, and he has an injury that could be potentially affecting his mind and/or faculties). It could be very well possible that Gabriel Lorca is a cover name since there's many suspicions of this ship being Section 31, or at the very least they all have cover names since their work is above Top Secret.
I also think the Romulans were involved too. That would make sense being what Kirk said, a long the lines. “Because of Garth, Spock and I are able to work together as brothers” It’s been established that the Romulans gave the cloaking technology to the Klingons. Due to how The Romulans interfere through proxy, it possible that they instigated in the shadows the Starfleet-Klingon war. Thinking it would weaken both sides, not realizing in the future that both sides would end up eventfully becoming allies. There is no doubt that the alliedship greatly reduced Romluan influence in the quadrants.
Gabriel Lorca could be Garth. He does fit the pattern of someone on the brink of a psychosis. Doing more research on the matter I found this information. The TOS episode Whom Gods Destroy was circa 2268. The founding of the Izar colony was circa 2183. That’s a difference of 85 years. The events of Discovery take place in 2256, so the events of Whom Gods Destroy is a deference of 12 years. That totally fits within the timeline of Garth becoming insane after the Battle of Axanar.
According to the Star Trek: Star Charts (pp. 32 & 36) and the Stellar Cartography: The Starfleet Reference Library ("Federation Historical Highlights, 2161-2385"), the M-class planet Izar was the third planet in the Izar (Epsilon Boötis) star system, located in the Alpha Quadrant. The primary of this system was a K-class star. in 2183, after the first manned landing by the SS Horizon, an Earth colony was founded on this planet. Within months of being founded, the Izar colony was admitted to the United Federation of Planets.
The dominant species were Humans. In 2370, the census counted 185 million living on this planet. The planetary capital was New Seattle. Major attractions on this planet included the Starfleet Tactical School and the Izar Institute of Meteorology. This world was a stop on the major space lanes of the late 24th century. In 2183, after the first manned landing by the USS Horizon, an Earth colony was founded on this planet. Within months of being founded, the Izar colony was admitted to the United Federation of Planets. The dominant species were Humans. In 2370, the census counted 185 million living on this planet. The planetary capital was New Seattle. Major attractions on this planet included the Starfleet Tactical School and the Izar Institute of Meteorology. This world was a stop on the major space lanes of the late 24th century
Also, I found it very interesting that the episode “Context is for Kings'” was spoken verbatim by Lorca within a context that would seem borderline maniacal, if not unethical. I think we're seeing the so-called "Origin Story" of Garth of Izar.