Now if that character -that did SO many things for all those key-figures of the Third Age- did not exist, or at least Tolkien didn't mention, why is it so bad to have them fight a battle that is SO EPIC?
Because it's not their fight, not this time. This isn't filling in some gap, it'd be a clear break with the story. The expectation to be involved in every damn scene just because it's 'SO EPIC' is a bit much.
Anyway here's a good (I believe) trick that even our poor Dwarves, Elves and Hobbits participate in the battle :
It is known that among the defenders of Helm's Deep were old men and children.
Youths, not children. And given that hobbits were roughly the size of four-year-olds, the idea that anyone could mistake them for someone who ought to be running around on a battlefield isn't even the least bit plausible. Dwarves would stand out too, not just from their height and the beards but their build. Only Elves would have a fair chance of blending in.