Emulating distance on Poincaré disk for different curvatures

The distance $$d_K^H(x,y)$$ between two points on the hyperboloid $$H_K$$ with curvature $$K<0$$ can be emulated on the distance $$d_{-1}(x,y)$$ of the hyperboloid $$H_{-1}$$ of curvature ($$K=-1$$) as follows: $$d_K^H(x,y)=R\cdot d_{-1}^H(x/R,y/R)$$ where $$R$$ is the radius and is related to the curvature as follows: $$R=\frac{1}{\sqrt{-K}}$$.

Do you know about a simple formula to do a similar emulation with the distance $$d_K^D(x,y)$$ on the Poincaré disk $$D_K$$ of curvature $$K$$?

For $$K=-1$$ the distance on the Poincaré disk $$D_{-1}$$ is: $$d_{-1}^P(x,y) = arccosh\left( 1+\frac{2||x-y||_2^2}{(1-||x||^2_2)(1-||y||_2^2)} \right)$$ So I'm looking for an expression of the form: $$d_K^P(x,y)=\cdots d_1^P(\cdots x\cdots, \cdots y\cdots).$$ where the $$(\cdots)$$-parts are just replaced with some function or expression in terms of $$K$$ (or $$R$$).

So far I've tried to project the points from the hyperboloid to the Poincaré disk. But it didn't turn out to be a nice expression.

2. the scale of the model (for example, the radius of the Poincaré disk, or a parameter of the hyperboloid, or the radius of a sphere represented in the $$(x,y,z)$$ coordinates).
If you have a distance formula $$d(\overrightarrow{x})$$ for a model of radius $$R$$, then you can define a distance formula on a model of radius $$RC$$ with $$d'(\overrightarrow{x}) = d(\overrightarrow{x}/C)$$. The new model is clearly isometric to the old model, so this does not change the Gaussian curvature.
On the other hand, if you define $$d'(\overrightarrow{x}) = Cd(\overrightarrow{x})$$, this multiplies the radius of curvature by $$C$$. Just like on a sphere: if you take the distance formulas for a sphere of radius 1 (in whatever coordinates), and multiply them by 10, you get the distance formulas for a manifold isometric to a sphere of radius 10, and thus with a smaller curvature, but still parametrized as if it was a sphere of radius 1.
So the $$/R$$ part in your hyperboloid formula controls the scale of the model, and the $$R\cdot$$ part controls the Gaussian curvature. If you want a metric on the Poincaré disk of radius 1 that makes the curvature different, just multiply your formula by a constant. If you also want to change the radius of the Poincaré disk for some reason, then you have to also adjust the coordinates.