Fix a positive integer $m$ and let $\mathbb{Q} \cap (m, +\infty) = \{ r_k \}_{k \ge m}$. If $$A:= \left\{ m+r_m+\sum_{k=m}^n r_k : n \ge m-1\right\},$$ then we know that $A-A= \Bbb{Q} \setminus ([-m,m]\setminus \{0\})$.
But, we are looking for a subset $A$ of rationals such that $A-A= \Bbb{Q} \setminus \{ \pm1,\cdots,\pm m \}$ or (if it does not exist) $| \Bbb{Q} \setminus (A-A)|=2m$. Are there such subsets $A$?
Note that $A_i = \{0\} \cup \big((i,i+1)\cap \mathbb{Q}\big)$ satisfies $$A_i-A_i = \mathbb{Q} \cap \big((-i-1,-i)\cup (-1,1) \cup (i,i+1)\big)$$
Since your $A \subset [0,\infty),$ we can just union that $A$ with our $A_i$ shifted sufficiently far left to get an example. That is, take
$$A \cup \bigcup_{i=1}^{m-1}(A_i - 3mi)$$