Set-up:
Let $G$ be a finite group.
Let $R$ be a Dedekind domain of zero characteristic with $K=$ Frac$(R).$
Let $\mathcal{P}_R$ denote the category of finitely generated, projective $R[G]$-modules.
Let $\mathcal{L}_R$ denote the category of locally free $R[G]$-modules.
$\Big[$ An $R[G]$-module $X$ is called "locally free" if
$X$ is finitely generated over $R[G],$
$R_P\otimes_R X$ is free over $R_P[G]$ for all the maximal ideals of $R$ (where $R_P$ is completion at $P$).
Moreover, for $X \in \mathcal{L}_R$ we have that $K\otimes_R X$ is free over $K[G]$ and
$$\mathrm{rank}_{K[G]}(K\otimes_R X)=\mathrm{rank}_{R_P[G]}(R_P\otimes_R X)$$
for all maximal ideals $P$ of $R.$ We therefore have a group homomorphism $$\rho_R:K_0(\mathcal{L}_R)\to \mathbb{Z}; \; \rho_R: [X]\mapsto \mathrm{rank}_{K[G]}(K\otimes_R X),$$
where $K_0(\mathcal{A})$ denotes the Grothendieck group of an abelian category $\mathcal{A}.$ $\Big]$
Question:
In the literature, I have seen the "projective class group" of $R[G]$ defined both as
the kernel of $\rho_R,$
the kernel of the group homomorphism $K_0(\mathcal{P}_R)\to K_0(\mathcal{P}_K)$ given by $[P] \mapsto [K\otimes_R P].$
However, I am stuck to see why these are naturally isomorphic.
Any help gratefully received!