Let $f\in C[a,b]$, and let $r^*\in\mathcal R_{m,n}$ be a best uniform rational approximation represented by \begin{align*} r^*(x)=\frac{p^*(x)}{q^*(x)}. \end{align*} Assume that $p^*,q^*\in\mathbb R[x]$ are coprime, $\deg p^*=m$, $\deg q^*=n$, and $q^*(x)>0$ for every $x\in[a,b]$. Assume also that the linearised numerator space \begin{align*} \mathcal H_{r^*}:=\{u q^* - p^* v : u,v\in\mathbb R[x],\ \deg u\le m,\ \deg v\le n\} \end{align*} has dimension $m+n+1$ and is a Haar space on $[a,b]$. Then $r^*$ is characterised among sufficiently small type $(m,n)$ rational perturbations by the existence of points $a\le x_0<x_1<\cdots <x_{m+n+1}\le b$ such that \begin{align*} f(x_j)-r^*(x_j)=\sigma(-1)^j\|f-r^*\|_\infty, \qquad j=0,\ldots,m+n+1, \end{align*} for some sign $\sigma\in\{-1,1\}$.