He didn't. He got 3rd billing behind Cotten and Valli. It is interesting though that the film just feels like something Welles himself directed even though it wasn't. I guess the fact that he and Cotten were both in it?
Mitchum? He would have been just.odd in this role. Mitchum can do crazy psycho well (Night of the Hunter, Cape Fear) but Harry Lame is a charming scumbag sort of character and I'm not sure Mitchum is right for it. The character is a bit too verbose and articulate, though that might be Welles' own input.

It strikes me that after reading the script Coward and Mitchum might not have wanted to appear in a film where they'd have only ten minutes of face timemost of it in a Viennese sewerand speak only about 250 words. I've watched this great film half a dozen times since 1955 and can't figure out how Orson Welles got top billing for so little work. He must have had a good agent.

I can't see Mitchum being as charming as Welles, and the character needs to be charming.
I'd be afraid Mitchum would simply be "a bad guy", in which case the movie would not work. What makes to movie work so well is that harry (the "bad guy") is much more likable than Holly Martins (the "good guy").
That is certainly intentional, and not just an accident caused by the casting. Holly is ostensibly the protagonist of the story, but he is a loser and a sad-sack. Holly may
think
he is the hero, but that's just another example of his dopiness. Holly certainly thought Anna would see him as a heroic figure, and he was extremely wrong about that.