Is "fights his way down" natural if I want to emphasize that it's a struggle?

Yes, the word fight can also implies effort or struggle.

Here you have a deeper explenation of the word fight and his usage:

Cambdridge dictionary for Fight

Yes. "Fighting one's way" emphasizes a struggle, regardless of direction. (One could equally "fight one's way up a stairwell" with an equal amount of struggle implied, but would then be traveling in the opposite direction).

As Lambie states, the directionality of the fighting is immaterial: one can fight one's way up, down, in, out, etc.

However, I think properly parsing the sentence into its component parts may help you understand why it is grammatically correct. It is less "fights his way down" and more "fights his way (preposition indicating direction of the struggle)".

(Prepositional phrase describing location:Inside the burning building), (Subject: David) (predicate: fights his way) (prepositional phrase describing direction of action: down the smokey stairwell).

Thus, we can also substitute any other predicate denoting an action that has a directional component. For example "Inside the burning building, David walks/jumps/looks/sprays water down the smokey stairwell".