I am looking for a inner metric on $\mathbb{R}^2$ (that induces the standard topology) which is not Finsler.

By "Finsler" here I mean a metric that is obtained by the following construction:

1. pick a smooth structure on $\mathbb R^2$ and take a suitable continuous function $\mu$ on $T\mathbb R^2$
2. define a metric as $d(x,y):=\inf_\gamma \int \mu(\dot\gamma) dt $ over all piecewise smooth paths $\gamma$ connecting two points $x,y$.

If exists, it is necessarily non-invariant (Berestovski theorem).