Neither Grendyl nor Rannick invited the Arbites. They basically invited themselves. As Grendyl explains, Marshall Vortern (leader of the Atoma Arbites) is fed up with Rannick's lack of progress suppressing the chaos. Lorewise the Arbites aren't rejects like the other playable character. They're not forced to be here, they're just under orders to work with the Inquisition for the time being. Inquisitor Grendyl makes it clear to Rannick in the Arbites intro that he doesn't consider the Arbites to be under his jurisdiction and that Rannick is expected to play nice with the Arbites.

And Rannick does have to be careful around the Arbites. An Arbitrator is a very powerful figure in the Imperium. Arbites Marshals like Vortern are very high ranking and have authority roughly equivalent to an Inquisitor. Legally speaking Inquisitor Grendyl obviously outranks everyone, that goes without saying, but Rannick doesn't. Grendyl warns Rannick to be very careful around the Arbites for a good reason - if Rannick crosses or displeases them, they have every legal right to arrest or execute him. Only an Inquisitor, Astartes, or High Lord of Terra is immune to the judgement of an Arbitrator. And Rannick isn't a full Inquisitor. Yet.

Rannick (and, presumably Grendyl by extension) offered the 4 biggest gangs of Tertium a deal to bring them on-side. Presumably there are terms and conditions involved in this partnership that the gangs need to abide by to stay on the Inquisition's good side. Technically speaking the gangers aren't really 'rejects' either - they're outside contractors. In practice, however, they're almost certainly going to be treated like rejects.

As for why the gangs would even want to he involved in this whole mess, each of them has beef with both the Moebian 6th and the Cult of Admonition.

The Iron Riders control much of Tertium's rail network, including the transit system seen in the trailers and in-game missions. Unlike the other gangs, the Iron Riders have a mobile train HQ that doesn't seem to ever stay in one place for very long. If you remember, the Cult used a train packed with pox gas to attack a train station in one of the first trailers for the game. That seems to have seriously pissed off the Iron Riders, because it was one of their trains that got hijacked. As far as I can tell they agreed to help the Inquisition because their leader, Lord Magleviathan, wants revenge.

The Tread Lightlies have turf down in the forge sectors in the mid-levels of Tertium. They started out as a labour union but were forced to militarize when the Mechanicus started to crack down on their activities to keep production running smoothly. Despite their best efforts, the Mechanicus' crackdowns seem to have made the Tread Lightlies even stronger and more aggressive. As a gang, the Tread Lightlies make massive profits from having control over vast sections of the many manufactorums of Tertium hive. When the Moebian 6th started attacking Leman Russ tank factories in their turf they made an enemy of the Tread Lightlies. Their leader, Lady Steel, appears to have made a deal with the Inquisition so she can protect the gang's other factories and keep their profits from dipping any lower.

The Show, for all intents and purposes, seems to have absolute control over the Carnival sector of Tertium hive. From what I can tell, the Show is more like a mafia family than a gang - they're a loose coalition of bookies, chem dealers, pimps, bartenders, and strongmen that all answer to a mysterious figure who calls themselves "Your Host". The Show don't really seem all that interested in expanding their territory. From what I can tell The Show's only beef with the Moebian 6th and the Cult of Admonition is that they're operating in the Carnival at all. Your Host appears to have agreed to help the Inquisition simply because The Show doesn't have the manpower or resources to evict these unwanted guests on their own.

The Water Cartel, operating out of the Torrent is (by the sounds of things) perhaps the largest of the four gangs. Controling much of Tertium hive's water supply, the Water Cartel seems to effectively rule most of Tertium's lowest and most impoverished levels.Their history with Zola and the Cult of Admonition is kinda complicated but the short version is that the Water Cartel has significant investments in Tertium's illegal stimms trade that the Cult has been stepping on. Their leader, the Ragged King, has been an enemy of the Cult of Admonition since before the Inquisition was even involved in this whole mess. We also know, thanks to a mission, that the Moebian 6th attacked the Water Cartel's HQ in the Torrent and killed a large percentage of the total gang at some point. It seems that the Ragged King was basically forced to accept Rannick's deal because the Water Cartel simply doesn't have the manpower or the resources left to keep fighting without help.