Take $a_n=(-1)^n$. Then for $q$ even we have
$$ \frac{1}{N/q} \sum_{n \leq N \colon q\mid n} a_n - \frac{1}{N} \sum_{n\leq N }a_n =1 +o(1).$$
Squaring this and summing against $1/q$ we obtain $(\log Q)/2 +O(1)$. So the direct analogue of the large sieve inequality is false no matter how small $Q$ is.
The weaker statement may be true because this sequence is well-distributed to all odd moduli.