Does there exist $B \subset \mathbb{N}$ which is not an additive basis, but is such that for every set $A \subseteq \mathbb{N}$ of Schnirelmann density $\alpha$ and every $N$ there exists $b \in B$ such that $$\lvert (A \cup (A+b)) \cap \{1, \ldots, N\} \rvert \geq (\alpha + f(\alpha)) N$$ where $f(\alpha) > 0$ for $0 < \alpha < 1$?

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