The map does not have to be surjective.

Suppose that $N$ is divisible by $2$, but not by $4$. In this case, the group $\mathbb Z/N\mathbb Z$ has exactly one involution; hence, the dual group $\widehat{\mathbb Z/N\mathbb Z}$ has two real characters. Choose $\Gamma$ to consist of these two characters, and let $\delta\in(0,1)$. Then $\mathrm{Bohr}(\Gamma;\delta)$ contains $0$ and $N/2$, while all other elements of $\mathrm{Bohr}(\Gamma;\delta)$ come in pairs: $g\in\mathrm{Bohr}(\Gamma;\delta)$ if and only if $-g\in\mathrm{Bohr}(\Gamma;\delta)$. Therefore, $|\mathrm{Bohr}(\Gamma;\delta)|$ is even.

On the other hand, the intersection $\Lambda\cap[-\delta N/4,\delta N/4]^k$ has an odd size (it contains $0$ and is symmetric). It follows that $\mathrm{Bohr}(\Gamma;\delta)$ and $\Lambda\cap[-\delta N/4,\delta N/4]^k$ are not isomprphic.