There's something about Michael Franklin and (Trevor). With COL, I've struggled getting in their characters. Niko was easy, as was JK. Michael certainly has the deepest of characters but Franklin is just a shell. I tried to expand him, but it's been hard (writing so little in the last year or so doesn't help).
Gus Mota could have a unique story though. Ask yourself the question: what are you giving us with Michael and Franklin? How is it going to be different to GTA v? I praise your attitude to this, and wish you luck, and also don't wish to attack your confidence or enthusiasm (that you're doing it for fun is the most important thing; sometimes you do have to stand by your guns and say "no, im not changing it!"). My advice is to split this up:
Write Gus's backstory (I feel he'd be more interesting as you have a blank slate almost - look at Rami Yalon in JIF). Pad him out fully. Learn his character and his past - why he's doing what he is etc. Then do similar with your other characters. Then write their own plot lines - see my guide if need be for the 8-point plot arc - so know what their story is. Then think how they combine, and perhaps an overall plot. GTA v really is the story of Michael and Trevor - Franklin is just a counterweight, and imo that shows in his character. Remove Franklin and tbh the story wouldn't differ too
much. Remove Michael or Trevor and the story collapses. Make sure all of your protagonists are the latter; make sure their plot is so integral to that of the overall story that, like a load-bearing wall, it simply could not stand without it.
Knowing the characters is going to be key, so really take your time to do so; write dossiers, when your bored and alone, have conversations with them in your head, know how they'd react to almost anything. An example of this done well is Pulp Fiction - each character is its own and with strong development (from hunnybunny right up to Vega and Wallace, hell even the buddy holly waiter!). It's almost like writing three, four different stories that tie together. As you've already said: take your time, and be careful and attentive to your characters.
The reason I'm sceptical is simply the amount of work a single-protagonist story demands is easily quadrupled when using three/four characters. GTA is for example: niko's story* was superb - strong characters, strong subplots and an outstanding overall plot line. The DLC were still strong characters (perhaps made possible by shorter stories) and a story that does blend with the others, but it's all done well (admittedly, bogt being the 'weakest' but it's a different take on things)
Gtav, however feels like a watered down version. Weaker characters, weaker story**, and to me it's evident how they struggled to get to gtaiv/rdr's base narrative quality - rdr of course focused on one protagonist (jack was poorly done) with a cast of great characters that made a standard plot and excellent one
You'd do well to look at these, not as a gamer but as a writer - look at the narrative and plot lines, the characters and the themes and parallels. The same applies to films and of course books. You'll begin to see how the characters feel real (looking at rockstar and GTA games will also help with the GTA-style).
To do fan fiction well you have to be an expert on the game***. That's easy to do (play it a lot, read wikia etc). Hopefully by the time you post this, I'll have pulled my finger out and read carbonox and albanyave's works (among others) so I'll keep and eye out. Post here or in the writer's room for help or discussion if you want of course.
* I mean the narrative not the quality of the game
**this is of course my opinion
*** with original works you create and control the world; you can make add remove or change anything at any time. FF, though, it's already there, so you HAVE to get it right!