I would say it is doable if they have no more than one thing in common with the namesake character. For example, in Dan Brown's novel Digital Fortress, one prominant character is named Jabba... its his nickname and like the namesake, Jabba (hence forth Jabba II to distinguish him) is so named because he is obese and works in a computer science field that has a lot of nerdy friends who almost have all seen Star Wars more times than a reasonable human has any right to see them. Beyond that, he's not a naked slug who makes his living as a don in a mafia style organization. There's a difference between the more famous Jabba and the less famous Jabba II.
The points going against you are you're doing it multiple times and the strong possibility the relations between your characters and their namesakes correlates. It's not that theme naming isn't bad... it's quite common... but the relationships and the relative obscurity of the work is not helping stave off a suit... and even while there enough differences that they could be seen as the character, that's not going to stop a lawsuit from being filed against you. The goal of the lawsuit will be to get you to stop it... or financially ruin you trying to assert your right to do so (you don't get a free lawyer when you're hit with a civil suit). I would change this so that the reference pool is broad and not related to one medium or one company that works with that media. It's much more favorable if your one use from this series is counted among the likes of Kirk, Skywalker, and Frodo... companies like when there guy is being held to the obvious other character giants of the genre.
My problem here is that they all have nicknames that are very distinct (I can't think of another instance of any of them that I have ever seen in my life) and are given specifically for their role in the story... and they are all on a team that presumably needs those roles at some point? Now, I may have never played this game, but I'm pretty versed in the ways of copyright infringement for a person without a law degree, and my copyright senses are... hang on, I got to write a check to Mickey Mouse... tingling!
It be one thing if these were characters in a bit role... they exist in a small portion of the story... and even better helped if the names were close enough without gong over... for example, in my works I routinely use the names of the Animorph's main cast for students in the same class as my main characters that are tertiary to the story at best and I always use variant names of their names (i.e. Jake becomes Jacob, Rachel becomes Rachelle, Marco becomes Marcus, Cassie becomes Cassandra, Tobias becomes Toby, and Ax becomes Phillip (helps when you know that Ax is an alien and only uses Phillip when he's pretending to be human, and even then his interaction with humans who don't know he's an alien is kept limited because he's weird on his own.). They are not part of the story and could be named anything else but that's a quick grab bag of reliably diverse names I can pull from on the fly.
A good test is the following: IF CHANGING AT LEAST FOUR OF THOSE NAMES RUINS THE OVERALL PLOT OF YOUR STORY THAN STOP WHAT YOU ARE DOING RIGHT NOW. If that is not the case, than change the four names right now so that there is less offense (and keep the name of the guy who most needs it). Copyright law can get very nuanced very fast.
Also, don't bother reaching out to the company to ask their permission... they will tell you know because, hey, they would rather not have to pay their own lawyers for the headache you caused them.
ONE MORE THING... scribbles a quick check to Jackie Chan If you don't intend to make any money off of this story, take all my advice and throw it out the window... In order to prove copyright infringement happened, you have to show injury to the copyright holder and the brand. Very hard to do that when you didn't see any money from it.
So based on your comments, I'd like to run this suggestion by you. Since your reason for the theme naming is loyalty and comradery, there could be a way to preserve the theming but change the names to be less closely grouped. Heroes and villains alike place a lot of value on loyalty across fiction. Perhaps rather than pool your character names from one source, pool them from multiple sources and use characters you feel are exemplary of loyalty. This gives you some wiggle room to name characters across stuff, thus preventing them (especially since six in one group is hard to find... most groups tend to work in either twos, threes, or fives as a core with an occasional additional member who's inclusion messes up the established relationship dynamic). Another idea is to take the game names, and mix and match them with names of other characters who are similarly exemplify this. Also, I'd suggest that the acceptance of the nicknames on the team is varied between team members... perhaps one likes the idea but another is proud of his/her given name and outright refuses to go along with it. When you write, the former character can be addressed by the narrator by the nickname, but the latter character is addressed by the narrator by his given name (among the characters in dialog, who addresses him by which name can be mixed and varied... Consider in Star Trek: Deep Space Nine where the character of Major Kira Neirys was addressed by her sirname by almost everyone in the cast, but Odo, who was a personal friend and romantic interest called her by her given name (Kira's culture uses a Sirname Givenname format, so Kira is her Sirname. Also, that this was a development in the script as Kira outranked Odo, so he would still call her Kira when they were discussing official buisness... but called her Neirys to highlight that he's speaking as a friend and not a subordinate officer and even then, only really started once they became closer. Odo being all about order was prone to rules.).
Also, perhaps look into the nature of these names from the game and see where they derive from. Again, I haven't heard them, but if you can get a close but similar naming, it might be a good way for your nickname giver to be a little clever about it... he's preserving the name, but demonstrating he has scholarly command of the product by choosing names you'd have play the game or research the backstory of its creation to properly understand. This can allow you a bit more freedom with acknowledging the work because now you can discuss why the name is relevent to the team, which in turn informs the reader about some of the more intiment details about why you love the game (It doesn't have to be the nicknamer either... another member of the team could reveal he or she knows the name's significance because she's read his paper(s) on the work of art, which the other person has read prior to joining the team and showing unique insite into the name).
Speaking from personal experience, I do enjoy Shakespeare, but I never would have read the Bard if I hadn't first watch Gargoyles, which among other things, showed several of Shakespeare's characters as real characters that the Gargoyles could interact with (it's MacBeth character is actually closer to the Historical Scottish King than it is to the one featured in the play... whic not only makes the viewer want to go read the play, but also enjoy it because it's different enough to be a new story). It can also work if the name helps to show differences in the character from the homage character (why did I get this name? I don't represent it... Well, here's my thinking). One of my favorite Avatar: The Last Airbender episodes is the season two "Zuko Alone" which puts Zuko in the role of Shane from the classic Western film and novel of the same name. Here there are some differences... In Shane, Shane is not the viewpoint character of the story... a good chunck of it is about the other characters learning about who Shane is. In Avatar, Zuko is our viewpoint character and it's the farmers that take in Zuko who are. Avatar and Shane also feature a critical difference in that at the end, Shane hasn't revealed much of his past, with the few hints that we get being that it may not have been a savory one, but that doesn't matter because his actions in the present prove he's a good person. In Avatar, Zuko does reveal more, which results in a dramatically different reaction to essentially the same actions.
There are a lot of ways to go about this that take it from something the rights holder would sue over to something that the rights holder respects because it shows respect for the work... by widening the gap of the nicknames, you can better discuss your appreciation of the work because now it's respectful enough to be its own work and not a pale immitation.
There's a great video by an online reviewer called SFDebris that discusses the difference between homage and plagerism... his basic premise is that all works of fiction do not exist in a vacumme... its okay to borrow ideas from other works, as long as you impove upon them... and shows the chain of evolution in the popular phrase "The greatest trick the devil ever played was convincing the world he didn't exist" to highlight homage and plagerism.
Also check out the Doug Walker's video "Where's the Fair Use" (a.k.a. #WTFU and yes, the acronym was intentional) where he describes the living hell that people who are trying to protect their copyrights have put him under for his popular web series "Nostalgia Critic" as well as other internet reviewers even though under Fair Use laws, which cover works critical of other works, Walker and his other guests did nothing wrong... in one case, one guy never bothered to show a clip of the discussed work and got hit with copyright infringment charges. It's just another way to highlight that while this most likely intended as tribute, if a company doesn't see it that way, they can be extremely agressive in their defense, even when you are doing nothing wrong (they are banking on you being an up and coming author with nothing in terms of money forcing you to either cave or bankrupt yourself in legal fees trying to pass it off. In many cases, even if you prove your case that the work is not illegally using someone else's copyrights, the courts aren't automatically going to give you compensation for your effort unless it's so blatant an abuse of the law on the claimant's part). Keep in mind, when D.C. comics sued Fawcett Comics over the similarity between Superman (D.C.) and Captian Marvel (Fawcett) the case dragged on for a decade and Fawcett was under Cease and Desist for that entire period on what was at the time, the best selling title in the genre... and considering how profitable Superman has always been, that's a lot of money.