Giving a finite collection of infinite words "complex" enough with respect to automata measure
It is possible. Consider all languages $L_1,\dots,L_m$ recognizable by automata with at most $n$ states.

Now to each $x=x_1\dots x_m\in\{0,1\}^m$ we associate a (possibly empty) language $$L_x=\bigcap_{1\leq i\leq m} L_i^{x_i},$$ where $L^0$ is the complement of $L$ and $L^1=L$.

Let $Y=\{x\in\{0,1\}^m~|~ L_x\neq\emptyset\}$.
For each $x\in Y$, we choose $\xi_x\in L_x$. This gives a finite family $\{\xi_x~|~x\in Y\}$, we show that this family satisfies the wanted property.

Take any $\eta\in X^{\mathbb N}$, and for each $i\leq m$ let $y_i$ be defined as $0$ if $\eta\notin L_i$ and $1$ if $\eta\in L_i$.
We have $y=y_1y_2\dots y_m\in Y$, since nonemptiness of $L_y$ is witnessed by $\eta$. We show that $\xi_{y}$ satisfies the condition. Indeed, for any $L$ recognized by an automaton with at most $n$ states, $L$ does not separate $\eta$ from $\xi_y$, by definition of $\xi_y$.

Notice that this family is optimal in size, since if some signature $x\in Y$ is not represented by some $\xi_x$, the property fails. Its size is at most doubly exponential in $n$.