"This story isn't true."
I'm not sure what a marker particle is. I think 的 is used because 是 can't link an adjective to a noun and 的 creates something like a noun phrase/ nominalised adjective. 真的（故事） Other people may have a different opinion.
That's more or less what I meant by marker particle....the particle itself doesn't translate literally but serves as a marker to indicate something. In this case, maybe as you say, it nominalizes the adjective.
"真" is an adjective, so one needs "的" to make it behave like a noun because it's being linked via "是".
You can think of "這個故事不是真" as meaning "This story is not truly.", which is clearly wrong, and "這個故事不是真的" as meaning "This story is not a true one."