Trying to name a character without it being too much like other media's protagonists.

The witch, the huntress, the way, the spear, the final hope, the blood destined, the bane born, the drawn bow, the Victor, the designed, the focus, the promised arrow, the sworn sorrow, the vengeance, the reserved rage, the gore gilded, the solution, the purpose, the Vendetta, the last sight, the doom vision, the peril, the endless March, the blood heel, the blood fall...

I genuinely think you’re fine to use Slayer or Berserker or anything else you want! Those terms aren’t copyrighted by any means and I see them pop up all the time in literature. I mean, Berserkers we’re a type of Viking warrior. And if Slayer was taken by Doom, the world never would’ve gotten Buffy! :)

Depending on the size of your world and how info is shared, it would be cool if your protag had many titles depending on the culture where they’re known. Like, a Norse inspired culture might call the protag “the berserker”, someone else might call them “The Warrior,” maybe others call them “the Slayer,” or even “the savior” or “the saint” etc.

In my head she's called 'The Slayer' but I think it's a bit too close to DOOM, the game.

Funny, my first thought was Buffy the Vampire Slayer, since she's often referred to as the Slayer.

Maybe something to compare them to like a mythical creature

“The Wyverns” for their stealth, their speed, and their terrifying ability to sly their target

The reason I’m thinking from this perspective is like how John Wick was nicnamed the Babayaga

Depending on the seriousness of the story, you could have the creators of this being call it something absurdist like, "The Jim." And then backronym your way into something more serious, like, Juggernaut Immortal Mangler.

The Defiler

If it's just legal stuff you're worrying about, then don't! Unless your slayer is a demon hunting, Gun toting, green helmet wearing hell spawn then your fine

Outside of copyright I find it a little generic and corny though But maybe that's just me or maybe that's what your looking for

The Chosen One, for extra corniness.

Jokes aside, maybe go for The Sin-Eater, The Devourer, The Herald, The Shade of Vali (Vali is a norse god born only to avenge his half brother’s death), The Debt-Collector, The Catchpole, The Night Butcher, The Venatrix (Latin: Huntress), The Calamity, The Unshealthed, The Executioner, The Wrath

People are often too critical pointing out similarities in fiction. If this is what you want to do, then do it. Your work and names is always going to be compared, but in the end, tell your story and they can tell theirs.

Originality is always important but in trying to avoid criticism we become what we fear most, someone who conforms completely to others' opinions.

I know its not that deep, but from personal experience, the best way to not drive yourself mad is to quite worrying about similarities and write the story you want to tell.