Your metric has to be geodesic, in particular $$d(x,y)=\min \{\,1,|x-y|\,\}$$ is not Finsler in your sense.
Now let $d$ be a geodesic metric on $\mathbb{R}^n$.
Suppose $n=2$. In "Two counterexamples..." by Burago, Ivanov, and Shoenthal, it was conjectured that a neighborhood of any point in $(\mathbb{R}^2,d)$ admits a Lipschitz embedding into the Euclidean plane.
Suppose $d$ is a Finsler metric in your sense. Since $\phi$ is continuous, the natural map $(\mathbb{R}^n,d)\to \mathbb{R}^n$ is a locally lipschitz homeomorphism. So this conjecture is closely related to your question.
Now suppose $n=3$. The same paper provides an example of a metric $d$ on $\mathbb{R}^3$ that (locally) does not admit a Lipschitz embedding into $\mathbb{R}^3$. (The construction is interesting --- it is worth reading.)
