Let $\mathbb{H}$ denote the upper half-plane with hyperbolic metric. Choose $p \in \mathbb{H}$. Let $\delta:=\lim\limits_{R\to \infty}\frac{\log(Vol\left(B(p,R)\right))}{R}$ be the volume entropy. This is independent of base point and a calculation shows $\delta=1$.
Question: Is it true that there exists a positive constant $c$ having the property:
For any $2<R, 0<r<1 $, one can find $k > c\frac{\exp(\delta R)}{r}$ points $p_1,\dots,p_k$ with $d(p_i,p)=R$ and $d(p_i,p_j)>2r$?
All I see is that $\exp(\delta R)$ will tend to $Vol(B(p,R))$ as $R$ tends to $\infty$.
Upper half-plane model is not relevant here -- this is a problem about the hyperbolic plane, you do not use the specific model in the statement, and I do not think it provides any insight about how to solve the problem. (I would use the Minkowski hyperboloid model myself.)
So you have a circle of radius $R$ and want to fit points in distance $2r$. An almost solution is to note that a hyperbolic circle of radius $R$ has length $2\pi \sinh(R)$, so you would fit $\frac{2\pi \sinh(R)}{2r} = \Theta(e^R/r)$ points on it.
The catch is that this measures distances along the circle -- but if the distance along the circle is $2r$, the actual distance between two points, i.e., the chord, will be shorter (they will be almost the same if $r$ is small, but the ratio is larger when $r$ is large and $R$ is small). So you would have to show that the ratio of distance along the circle to the chord is bounded by a constant when $2<R, 0<r<1$. I have not computed this, but it should be easy to do e.g. by using the hyperbolic version of the cosine theorem to compute the angle (as seen from $p$) between two points with distance $d$ along the chord.