I think we do have three very good arguments why we should be allowed to tell the story from the book:
1. Eol would only claim he has more right to live in Beleriand than the Noldor do if he arrives before they do, not years afterwards. It's a significant part of his story and character that he was there first, years before the Noldor, and that he enjoyed peace in Beleriand and blamed the Noldor for starting the war with Morgoth -- not just that he has a generic, unspecific dislike of them for no reason. And that he got to know the Dwarves long before they met any Noldor. If he hates the Noldor for no reason at all, then he'll be like a characature of the original character. Eol in the book is a nasty person, but he actually has reasons for what he thinks and does -- they're generally bad reasons, but they make sense to him and he can tell them to other people -- he doesn't just hate people at random. If it was just reflexive xenophobia or being antisocial, he'd hate the Dwarves, too.
2. His choice to live in Nan Elmoth was also supposed to be about finding someplace safe in Beleriand, but outside the Girdle of Melian. That doesn't matter if the Feanorians have already killed all the Orcs, the Girdle is a decade old, and the Spiders are long gone, by the time Eol shows up -- he could live anywhere at all, without paying for it with magic swords, so why would he bother to pay for Nan Elmoth? Why would he choose that place, after the Noldor arrive? If he hates them, he wouldn't even emigrate into Beleriand, after it's settled by Noldor and he has to pass through Feanorian territory just to get to Nan Elmoth. How would he even know what Nan Elmoth is, or speak Sindarin, or know to ask Thingol's permission, if he had no prior contact with the Sindar?
3. How would we depict him hating the Sun and Moon and longing for the ancient time of starlight, if he doesn't show up until after the Sunrise? His unhappiness wouldn't really have meaning without contrasting it to a previous time when we show him to actually be content or happy, before the Sun rises.
None of the above would make sense if he arrived after the Noldor were already established in Middle-earth. Making him a late-comer to Beleriand instead of someone who lived there for many years fundamentally changes who the character is.
If the Hosts dislike flashbacks that much, then I wish they would allow us to tell the stories in chronological order, in the first place.