Okay. I think we can do this without making Fingolfin a whiny teenager. After all, he has adult children himself at this point.
Beat one: Fëanor is obnoxious and insufferable...because of course he is. We show him saying or doing something that is very dismissive of his younger half-brothers. It could be as simple as one of his sons saying, "So, when are we going to visit our cousins?" and Fëanor responding, "You mean your half-cousins? You know I am my parents' only child." or "Neither of them has ever made anything that was worth any notice - it's as if they aren't even really Noldor." or something like that. Fingolfin either overhears this directly, or his children complain to him of their uncle being an insufferable jerk. We see that Fingolfin has done a lot of putting up with and suffering in silence through these slights throughout his entire life.
Beat two: We show anti-Indis gossip in Tirion. Melkor is clearly involved in this in some way, but it doesn't seem to come from him. If he does say it directly, it is to repeat it (reluctantly) to Fingolfin, but better it just be 'word-on-the-street'. The point of this is to show that Fëanor is not the only one who is dismissive of Indis' children, and that Fëanor's slights will find a ready audience for repeating them.
Beat three: Melkor-the-fortune-teller calls out Fingolfin as a king, or even heir of Finwë (High King of the Noldor). When word of this reaches Fëanor (via Caranthir), he retaliates in some way (refuses to see/speak/visit with him?).
Beat four: Fingolfin has had enough. He ignores his mother's advice to continue placating Fëanor, and starts telling people how he really feels about the idiotic behavior of his older (half-)brother. Finarfin urges peace, and Fingolfin retorts with something along the lines of, 'What has that ever gotten us?'
Beat five: Melkor lets slip about the Second Children. This news sends a wave through the Noldor community and is *the* news. Fëanor reacts badly. Fingolfin attempts to be the voice of reason. Fëanor does not flat-out blame or accuse the Valar of anything, but certainly insinuates that trusting them to be fully honest is a mistake. Fingolfin defends the Valar. These conversations are separate; they aren't talking to each other.
Beat six: Fingolfin and Fëanor have an argument-by-proxy with each other about the Valar and the Second Children. It is clear that they are fighting about this because they are really fighting about their family. Finarfin as proxy?
Beat seven: Fëanor, in full on paranoia mode, denounces his half-brothers as usurpers, sycophants, idiots, leeches, and whatever other derogatory thing he can think of. Thanks to his sons, these thoughts are not kept private, but are very much the gossip of Tirion. Responses are put into Fingolfin's mouth, and find their way back to Fëanor. Melkor, of course, is sitting back and watching the show and playing telephone to twist what is said even more - there's a grain of truth in all of it, but it's so easy to make it much worse than it is, and since they aren't talking to each other - no way to uncover the truth.
Beat eight: Fingolfin, having had enough, seeks out Finwë in a private audience to call him on doing nothing while his sons fight publicly. Finarfin does not go with him, but word reaches Fëanor that his brothers have gone to the palace. He shows up in full on regalia, and accuses Fingolfin of carrying lies to their father behind his back.
Beat nine: Fëanor draws his sword on Fingolfin after he leaves the audience with their father.
Beat ten: Fingolfin leaves without saying anything and speaks with Finarfin about this latest over-the-top insult. Something is said about 'Why must everything Fëanor does be taken to such extremes?'
Beat eleven: The Valar exile Fëanor from Tirion, announcing this through Eönwë.
Does that work?