Let $a:=\mu_1(0)$ and $b:=\mu_n(1)$. The linear span of the $\{f_k\}_{k\in\mathbb{N}}$ is  $\big\{\sum_{j=1}^nP\circ\mu_j  : P\in\mathbb{R}[x]\big\}$, whose closure in $C^0([0,1]),\|\cdot\|_\infty$ contains the space $\big\{\sum_{j=1}^nf\circ\mu_j  : f\in C^0([a,b])\big\}$, just because polynomials are uniformly dense in $C^0([a,b]) $ and $f\mapsto f\circ\mu_j$ are (linear) continuous maps. So the question is: can any $\alpha\in C^0([0,1])$ be written in the form $$\alpha=\sum_{j=1}^nf\circ\mu_j$$ for some $f\in C^0([a,b]) $? 


Let's define inductively a sequence in $[a,b]$ putting $c_0:=\mu_n(0)$ and $c_{k+1}:=\mu_n(\mu_{n-1}^{-1}(c_k))$ until we reach some $c_K$ out of the range of $\mu_{n-1}$, which happens in finitely many steps, because $$c_{k+1}-c_k=\mu_n(\mu_{n-1}^{-1}(c_k))-\mu_{n-1}(\mu_{n-1}^{-1}(c_k))\ge\delta:=\min_{0\ge x\ge1}\mu_n(x)-\mu_{n-1}(x)>0.$$
Since $\mu_n:[0,1]\to[c_0,b]$ is invertible, we can define arbitrarily $f$ on $[a,c_0]$, and state the functional equation for $f$ on $[c_0,b]$ equivalently as:
$$f(y) =\alpha(\mu_n^{-1}(y))-\sum_{j=1}^{n-1}f(\mu_j(\mu_n^{-1}(y)),\qquad y\in[c_0,b].$$
But this equation is self-solving: the RHS gives the unique extension of $f$ to  the interval $[a,c_{1}]$, then to $[a,c_{2}]$, till we cover $[a,b]$.