Let $\sigma$ be a continuous unitary representation of the topological group $G$ on a Hilbert space $V$. Suppose $\sigma$ weakly contains the trivial representation, that is: for any compact subset $K$ of $G$ and any $\epsilon > 0$, $V$ contains some $(K, \epsilon)$-invariant vector $v$ of norm 1 (i.e., such that for all $g \in K$ we have $|| \sigma(g)v - v|| < \epsilon$).
Is it true that for all $K, \epsilon$ there exists some irreducible representation $\rho \subseteq \sigma$ with $(K, \epsilon)$-invariant vectors? Would it help to add some hypothesis on $G$ (apart from abelian)?
I have changed my comment into an answer, as it contained something incorrect and something not fleshed out. If $G$ is compact, then by the Peter-Weyl theorem, $V=\bigoplus_iV_i$ is a Hilbert space direct sum of finite-dimensional irreducible representations. If $v$ is $(K,\epsilon)$-invariant and $v_i\in V_i$ is the nonzero component of $v$ in $V_i$, then by orthogonality of the direct sum, for all $g\in K$, $$ \sum_i\|\sigma_i(g)v_i-v_i\|^2=\|\sigma(g)v-v\|^2\leq\epsilon^2\|v\|=\sum_i(\epsilon\|v_i\|)^2. $$ It follows that some $v_i$ must be $(K,\epsilon)$-invariant.
For more general $G$ I don't know, but this seems related to Kazhdan's property (T).