I think that Lennie's fate is established in the first chapter of the book. George and Lennie are about to go to sleep and George reminds Lennie of what he needs to do if he winds up in any trouble:
“'Course you did. Well, look. Lennie—if you jus’ happen to get in trouble like you always done before, I want you to come right here an’ hide in the brush.”
Lennie's fate is one where he is predisposed to finding himself in difficult situations. He is going to be condemned to live a life where trouble always visits him. Lennie is incapable of living a life where trouble is not evident. George's "escape plan" is something that Steinbeck uses to foreshadow this. In the remarkable symmetry of the book, what is evident in the beginning comes back to the end. When Lennie kills Curley's wife and it dawns on him that he "done a bad thing," the first thing that pops into his mind is to go ahead and follow George's advice. In telling himself that he needs to "hide in the brush," it is clear that Lennie's fate has been decided. George's advice is the only thing Lennie remembers in the most critical and most forlorn of moments. It is this condition that seals Lennie's fate. George arrives and kills the unsuspecting Lennie, knowing that his fate at the hands of the men searching for him would be much worse that his painless death at George's hands.