The answer is that the above is equivalent to non amenability.
Fix a group $(G,*)$.
Since $(G,*)$ is non amenable if and only if every finitely generated subgroup is non amenable,
we may assume that $G$ is finitely generated.

If $\mu$ and $\nu$ are finitely supported probability measures on $G$,
define
$$
\mu * \nu (Z) = \sum_{x * y \in Z} \mu (\{x\}) \nu (\{y\})
$$
Observe that $g * \nu (E) = \nu ( g^{-1} * E)$.
If $S$ is a subset of $S$, let $P(S)$ denote all probability measures on $S$
(which are identified with probability measures on $G$ which are supported on $S$).
I will identify $G$ with the point masses in $P(G)$.

If $A$ and $B$ are subsets of $G$ and $A$ is finite, we say that
$B$ is $\epsilon$-Ramsey with respect to $A$ if for every $E \subseteq B$,
then there is a $\nu$ in $P(B)$ such that $P(A) * \nu \subseteq P(B)$ and
$$
|\mu * \nu (E) - \nu (E)| < \epsilon
$$
for all $\mu$ in $P(A)$.
Notice that in some sense $E$ is defining a partition of $P(B)$ and we are
postulating the existence of a copy of $P(A)$ in $P(B)$ which is homogeneous for $E$ up to
an error of $\epsilon$.

It can be shown with an argument similar to the one below that if $B$ is $\epsilon$-Ramsey with respect to $A$, then for every $f:B \to [0,1]$
there is a $\nu$ in $P(B)$ such that
$$
|f(\mu * \nu) - f(\nu)| < \epsilon
$$
where $f$ has been extended linearly to $P(B)$.

We say that $(G,*)$ is Ramsey if for every finite subset $A \subseteq G$ and every $\epsilon > 0$,
there is a finite subset $B$ of $G$ with is $\epsilon$-Ramsey with respect to $A$.
Notice that if $B$ satisfies that for every $E \subseteq B$ there is a $\nu$ in $P(B)$ such that
$$
|g * \nu (E) - \nu (E)| < \epsilon
$$
for all $g$ in $A$, then $B$ is contained in a finite set which is $\epsilon$-Ramsey
(we need only to replace $B$ by $A * B \cup B$).

To connect this to the question, suppose that $G$ is not Ramsey, as witnessed by a finite
$A \subseteq G$ and $\epsilon > 0$.
I claim there is a set $E \subseteq G$ such that for every $\mu \in P(G)$, there is a $g \in A$
such that $|\mu(E \cdot g) - \mu (E)| \geq \epsilon/2$.
Let $B_n$ $(n < \infty)$ be an increasing sequence of finite sets covering $G$.
Let $T_n$ be the set of all subsets $E$ of $B_n$ which witness that
$B_n$ is not $\epsilon$-Ramsey with respect to $A$.
Observe that if $E$ is in $T_{n+1}$, then $E \cap B_n$ is in $T_n$.
Otherwise there would be a $\nu$ in $P(B_n)$ such that $g * \nu$ is in $P(B_n)$ for each
$g$ in $A$ and
$$
|g * \nu (E \cap B_n) - \nu (E \cap B_n)| < \epsilon
$$
Such a $\nu$ would also witness that $E$ is not in $T_{n+1}$.
Define $T = \bigcup_n T_n$ and order $E \leq_T E'$ if $E = E' \cap B_m$ where $E$ is in $T_n$.
This order makes $T$ into an infinite finitely branching tree.
By König's lemma, $T$ has an infinite path whose union is
some $E \subseteq G$.
If there were a measure $\mu$ which was $\epsilon/2$-invariant for $E$ with respect to translates by elements of $A$,
there would be a finitely supported $\nu$ which was $\epsilon$-invariant for $E$ with respect to translates
in $A$.
But this would be a contradiction since then the support of $\nu$ would be contained in some $B_n$
and $\nu$ would witness that $E \cap B_n$ was not in $T$.

Now the claim is that the Ramsey property of a discrete group is equivalent to its amenability.
That amenability implies the Ramsey property follows from Følner's characterization of amenability.
Also observe that $G$ is amenable provided that for every $\epsilon > 0$, every
finite list $E_i$ $(i < n)$ of subsets of $G$, and
$g_i$ $(i < n)$ in $G$, there is a finitely supported $\mu$ such that
$$
|\mu (g_i * E_i) - \mu (E_i) | < \epsilon.
$$
Set $B_{-1} = \{1_G\} \cup \{g^{-1}_i :i < n\}$ and construct a sequence
$B_i$ $(i < n)$ such that $B_{i+1}$ is $\epsilon/2$-Ramsey with respect to $B_i$.

Now inductively construct $\nu_i$ $(i < n)$ by downward recursion on $i$.
If $\nu_j$ $(i < j)$ has been constructed, let $\nu_i \in P(B_i)$ be such that
$$
|\mu * \nu_{i} * \ldots * \nu_{n-1} (E_i) - \nu_i * \ldots * \nu_{n-1} (E_i)| < \epsilon/2
$$
for all $\mu$ in $P(B_{i-1})$.
Set $\mu = \nu_0 * \ldots * \nu_{n-1}$.
If $i < n$, then since $\nu_0 * \ldots * \nu_{i-1}$ and
$g_i^{-1} * \nu_0 * \ldots * \nu_{i-1}$ are in $P(B_{i-1})$,
$$
|g_i^{-1} * \mu (E_i) - \nu_i * \ldots * \nu_{n-1} (E_i)| < \epsilon/2
$$
$$
|\mu (E_i) - \nu_i * \ldots * \nu_{n-1} (E_i)| < \epsilon/2
$$
and therefore
$|\mu (g_i * E_i) - \mu (E_i)| < \epsilon$.