Your definition of "hugely unlikely" is... strange. Until you see how Turbine chooses to integrate our characters into the fight, it's premature at best to opine on how much they would have "stuck out like sore thumbs," or otherwise been remarkable enough to have made it into the account of the Red Book.
What, because Dwarves and hobbits wouldn't stand out among all those big, strapping Northmen types who live in Rohan? And I seem to remember that the player-characters are supposed to be heroes (super-heroes, really, given the plot of the game) but you're now trying to imply that at this one battle they're going to do so little of note that nobody will even notice them.
To call this unconvincing would be putting it mildly.
Further, it's somewhat silly to talk about "what sort of fantasy it is natively" as if to suggest that some arcane assignment of a fantasy subgenre dictates what could or could not happen in its world, given that Tolkien predates the parceling of fantasy into subgenres, as well as predating most of fantasy as a genre altogether.
The sort of OTT heroic fantasy the game engages in actually predates LOTR and even The Hobbit, it's got far more in common with 'pulp' fantasy written by the likes of Robert E. Howard. The trope of the solo hero overcoming all obstacles is notably lacking in LOTR, in case you hadn't noticed. Likewise the casual flinging about of magic.
If you want to take it on its face, it's no kind of fantasy at all, but legend, and one of the defining aspects of legend is to redraw as larger-than-life the formative events of an earlier age.
It's constructed legend, which is a type of fantasy.
But, let's keep saying it: absence of evidence is not evidence of absence. This isn't a blank check, but please be more creative than this particular (fallacious) train of thought.
You've not shown it to be fallacious, and the only creative thing I'm seeing from you is in the manufacture of one excuse after another. Take what you said earlier:
Tolkien was not specific (within LOTR) about the Iron Garrison, Rune-Keepers, the Golden Host, Draigoch, or the specific and precise whereabouts of the Lieutenant of DG throughout the events of LOTR. He wasn't. I've read the books. That stuff is definitely "in the gaps" stuff.
Not even close: those things cannot be said to simply fill gaps in the story because it is beyond credibility that they could have happened and not had consequences, nor passed unnoticed. Diverting resources away from Erebor when it was about to be attacked by Sauron? That would have had consequences. We know the Galadhrim did NOT attack Dol Guldur until after Sauron's fall, because there's a clear if very brief account of what happened in LOTR itself so that one is a straightforward (and significant) change. RKs are simply not credible (even if the idea behind them was even self-consistent, which it isn't, much less consistent with Tolkien's take on magic) because their existence could hardly have passed unnoticed for thousands of years, yet accounts of Dwarves in battle simply (and only, and always) have them going all medieval on the foe, not using any sort of magic (other than that in their weapons and armour). Draigoch? What, everyone just forgot to mention a dragon, of all things? People do tend to remember dragons, you know. And the people who kill them. Stuff of legend and all that. (Especially when they're so far off the beaten track for dragons, too). Real filling-in of gaps would be relatively low-key derring-do that might feasibly pass unnoticed among the general mayhem, not all this massively OTT epic stuff that Turbine are so fond of because it's exactly the sort of thing that people would tell stories about.
You can't have continual epic goings-on and then try to pretend everyone forgets everything about them every single time. They're changes to the story, not things that fit neatly into gaps. There's nothing wrong with such changes in principle but to try to pretend they're not changes is disingenuous at best. And that's before I point out what should be the obvious fact that the player-characters couldn't 'really' have been involved in all those events because there simply wouldn't be enough time to do all the travelling that would be involved, at Middle-earth's true scale. The game's story is a fudge, which should make it even more obvious that you shouldn't try to treat it as something that dovetails neatly with the books.