Since the norm is quite specific, I am not sure if you can find it in any book. However, you can prove compactness of the embedding directly. Given a sequence $f_k\in C^{\beta}(\mathbb{R}^n)$, the compactness for bounded domains and a standard diagonal argument shows that you can find a subsequence $f_{k_\ell}$ that converges to some $f$ in $C^\alpha$ on any ball $B^n(0,R)$. Then it is easy to prove that this subsequence converges to $f$ in $C^{\alpha,-\delta}$ because roughly speaking: on a large ball the $C^\alpha$ norm of $f-f_{k_\ell}$ is small and on the complement on a large ball the $C^{\alpha,-\delta}$ norm is small.

Since the norm is quite specific, I am not sure if you can find it in any book. However, you can prove compactness of the embedding directly. Given a sequence $f_k\in C^{\beta}(\mathbb{R}^n)$, the compactness for bounded domains and a standard diagonal argument shows that you can find a subsequence $f_{k_\ell}$ that converges to some $f$ in $C^\alpha$ on any ball $\mathbb{B}^n(0,R)$. Then it is easy to prove that this subsequence converges to $f$ in $C^{\alpha,-\delta}$ because roughly speaking: on a large ball the $C^\alpha$ norm of $f-f_{k_\ell}$ is small and on the complement on a large ball the $C^{\alpha,-\delta}$ norm is small.