The fact that the authors of https://arxiv.org/pdf/math/0602479.pdf refer to using "representation (4) for the distance" makes me think that the estimate you want is supposed to come from something along the lines of:
$$ \begin{array}{lllll} \Vert Df(x)\Vert &=& \limsup_{y\to x}\dfrac{|f(x)-f(y)|}{\Vert x-y\Vert}\\ &\leq & \limsup_{y\to x} \dfrac{d(x,y)}{\Vert x-y\Vert} \\ &\leq & (\delta^{-1}+\beta)\limsup_{y\to x} \dfrac{\rho(x,y)}{\Vert x-y\Vert} \\ &=& (\delta^{-1}+\beta)v(x) \end{array} $$
using (4) and $\mathrm{Lip}_d(f)\leq 1$ to get from the first line to the second, the second display on page 16 to get from the second to the third, and then waving your hands a bit to argue that
$$ \limsup_{y\to x}\dfrac{\rho(x,y)}{\Vert x-y\Vert}= v(x) $$
I might be missing something, but I don't think your (1), (2) and (3) are relevant for the derivative estimate.