To get started: If $c:[0,1]\to E$ is a $C^1$-curve from $x$ to $y$ you have
$$ |f(x)-f(y)|=\left|\int_0^1f\circ c(t)dt\right|\le \int_0^1 \|Df(c(t))\|_{E'}\|c'(t)\|dt \le \sup\{\|Df(z)\|_{E'}/v^r(z):z\in E\} \varrho(x,y).$$
This gives you an inequality for (4) (I guess that $x$ is fixed, there). For the other inequality I would try to take a direction $r\in E$ which (almost) maximizes $|Df(x)(r)|$ and take $y=x+\varepsilon r$. However, I did not try to work this out.