While trying to get an analogue of Weiss's monotiling result for amenable residually finite groups in the topological setting, I face the following problem.

Let $G$ be a locally compact amenable group with Haar measure $\mu$ having the following property (some kind of residual compactness or residual systolicity, see also A generalization of residual finiteness to topological groups): For every compact $L \subseteq G$ there is a cocompact lattice $\Lambda$ such that $L \cap \Lambda = \{1\}$.

Is the following true?

For every compact $K \subseteq G$ and $\epsilon>0$ there is a compact subset $F \subseteq G$ with positive measure, a cocompact lattice $\Lambda \leq G$ and a measurable section $\xi: G/\Lambda \to G$ such that

(1) $\pi: G \to G/\Lambda$ is injective on $K \cup FK^{-1}$,

(2) $\nu(\{a \leq G/\Lambda : \xi(a) \pi(K) \cap \pi(F) = \emptyset \}) \geq \nu(G / \Lambda) - (1+\epsilon) \nu(\pi(F))$ where $\nu$ is the induced Haar measure on $G/\Lambda$.

We can always fulfill (1) by our residual compactness assumption. The existence of a measurable section is ensured e.g. by Theorem 1 from http://projecteuclid.org/euclid.pjm/1102986142 (Feldman and Greenleaf. 1968. “Existence of Borel Transversals in Groups.”)

Since $G$ is amenable, we can also choose $F$ $(K^{-1},\epsilon)$-almost invariant in the sense that $\mu(FK^{-1}) \leq (1-\epsilon) \mu(F)$. If we could find a lattice $\Lambda$ fulfilling (1) that would furthermore be a normal subgroup, we would also have (2) by \begin{align*} \nu(\{a \in G/\Lambda : \xi(a) \pi(K) \cap \pi(F) = \emptyset \}) &= \nu(\{a \in G/\Lambda : a \pi(K) \cap \pi(F) = \emptyset \}) \\ &= \nu(\{a \leq G/\Lambda : a \not\in \pi(FK^{-1}) \}) \\ &= \nu(G/\Lambda) - \nu(\pi(FK^{-1})) \\ &= \nu(G/\Lambda) - \mu(FK^{-1})\\ &= \nu(G/\Lambda) - (1+\epsilon) \nu(\pi(F)). \end{align*} But the existence of such a normal $\Lambda$ seems to be to strong as can be seen by the comments in A generalization of residual finiteness to topological groups.