Using the fact that the only type of discontinuity compatible with monotonic function is the jump discontinuity, I showed that $A:=\{a\in[0,1]:f\textrm{ is discontinuous at }a\}$ is enumerable for every $f\in X$.
But that didn't help, once for a fixed $f\in X$, $\pi^{-1}[]f(x)-\epsilon,f(x)+\epsilon[]$ is an open nhood of $f$ for every $x\in[0,1]$. I cannot see how a enumerable family of open nhoods of $f$ can be a local basis in this case.
Let $f\in X$ be an arbitrary function, $S=\{a_n\}$ be an enumerated countable set containing both the set $A$ of the discontinuity points of the function $f$ and a dense set $D\supset\{0,1\}$ of the segment $[0,1]$ (for instance, its rational points). We claim that the family $$U_n=\{g:\in X: |g(a_i)-f(a_i)|<1/n\mbox{ for each } 1\le i\le n \}$$ is a countable base at $f$. It suffices to show that for any point $x\in [0,1]$ and any natural number $m$ the standard subbase neighborhood $V_x=\{g:\in X: |g(x)-f(x)|<1/m\}$ contains a set $U_n$ for some $n$. If $x$ is a discontinuity point of the function $f$ then the claim follows from the inclusion $S\ni x$. Conversely, there exists a number $\delta>0$ such that $|f(x’)-f(x)|<1/(5m)$ provided $|x’-x|<\delta$. Since the set $S$ is dense in $[0,1]$, there exist numbers $p$ and $q$ such that $$x-\delta<a_p\le x\le a_q<x+\delta.$$ Assume that $n>\max\{p,q, 5m\}$ and $g\in U_n$. Then
$$|g(x)-f(x)|\le |g(x)-g(a_q)|+|g(a_q)-f(a_q)|+ |f(a_q)-f(x)| \le$$ $$|g(a_p)-g(a_q)|+|g(a_q)-f(a_q)|+ |f(a_q)-f(x)| \le$$ $$|g(a_p)-f(a_p)|+ |f(a_p)-f(a_q)|+ |f(a_q)-g(a_q)|+|g(a_q)-f(a_q)|+ |f(a_q)-f(x)|<$$ $$\frac 1{5m}+\frac 1{5m}+\frac 1{5m}+\frac 1{5m}+\frac 1{5m}=\frac 1m.$$
Thus $g\in V_x$.