Is there a W[1]-hard problem that can be solved in $2^{o(n)}$ time?
How about Planar Capacitated Dominating Set? It is W[1]-hard (see the paper by Bodlaender, Lokshtanov, Penninkx in IWPEC 2009), but should be solvable in $2^{O(\sqrt{n}\log n)}$ by using the fact that planar graphs have treewidth $O(\sqrt{n})$.

In this problem we are given a graph with integer capacities on the vertices. We are asked to select a minimum size dominating set, with the added restriction that a selected vertex can dominate at most as many of its neighbors as its capacity. Even though the solution is not exactly a subset of $V$, I think it is known that, if we are given a supposed dominating set $D$ we can check if it can be turned into a valid (capacity-respecting) solution in polynomial time by reduction to Max Flow.