Let $X$ be a subset of $L^2([0,1])$ which contains only Lipschitz function.

Also, the member of $X$ are uniformly bounded
$$
|x(t)| < M, \text{ for all $x \in X$ and $t \in [0,1]$}.
$$
Let $F: X \to [a,b]$ be a bounded Lipschitz functional
$$
|F(x) - F(x')| \le \| x - x' \|, \text{ for all } x \in X.
$$
The minimum of $F$ can be approximate
by forming a finite $\epsilon$-net $D_m=\{x_i\}_{i=0}^m$ of $X$, and then taking
$$
\min \left\{F(x_i) : x_i \in D_m \right\} , \tag{1}
$$
as an approximation to $\min\left\{ F(x) : x \in X \right\}$.

What is a good way to construct the $\epsilon$-net $D_m$?

Since the function in $X$ are Lipschitz, I would use truncated Fourier series of order $n$ to build $D_m$.

Is it then true that for any $x \in X$, there is an $x_i \in D_m$ for which
$$
\|x - x_i \| \le \frac1n? \tag{2}
$$
Or is (2) delivered only by using truncated Chebyshev polynomial of order $n$?

Are there other good choices for building $D_m$ than trigonometric polynomial?

My greatest fear is that $X$ is not compact.
So that no $\epsilon$-net can gives (1).

Is this fear justified?

From the different assumptions, I think it can be dispelled, but I don't see the proof.