This is a follow-up to this question. Basically, I would like to know whether the desired vanishing in question can be deduced if we are allowed to vary $G$.
Suppose $M$ is an oriented closed manifold. Fix a class $x\in H^*(M;\mathbb{Q})$. Suppose that for every group $G$, every class $y\in H^*(BG,\mathbb{Q})$, and every continuous map $$f\colon M\to BG,$$ we have that $$\langle x\cup f^*y,[M]\rangle=0,$$ where $[M]$ is the fundamental class in $H_*(M,\mathbb{Q})$.
Question: Does it follow that $x=0$?
Yes. Suppose for convenience that $M$ is connected. Then $M$ is equivalent to a classifying space $BG$ for $G$ a topological group equivalent to $\Omega M$. So we may take $f: M \to BG$ to be the identity map, and if $x \neq 0$, take $y$ to be dual to $x$ under the cup product pairing which is nonsingular.