Let $X$ be a rimcompact Tychonoff space, that is, a completely regular Hausdorff space with a base of open sets with compact boundaries. It is well known that $X$ has a maximal compactification with a totally disconnected remainder (see this paper of Jesús M. Domínguez) which is called the Freudenthal compactification of $X$ and is denoted by $\overline{X}$. Its remainder is denoted by Ends($X$) and it is widely known as the $\textbf{end space}$ of $X$ (see this question of mine for a more concrete description of the end space for proper geodesic metric spaces).
Now consider the algebras of real-valued bounded continuous functions
\begin{align*} C_{f}(X)&=\{f: X\rightarrow \mathbb{R}\mbox{ continuous}\hspace{0.1cm}|\hspace{0.1cm}\exists K\subseteq X \mbox{ compact such that }f(X\setminus K)\mbox{ is finite}\},\\ C_{c}(X)&=\{f: X\rightarrow \mathbb{R}\mbox{ continuous}\hspace{0.1cm}|\hspace{0.1cm}\exists K\subseteq X \mbox{ compact such that }f(x)=0\hspace{0.1cm}\forall x\not\in K\}. \end{align*}
According to Theorem 3.2 on the paper of Domínguez, every function $g\in C_{f}(X)$ can be continuously extended to a locally constant function $\overline{g}: \overline{X}\longrightarrow \mathbb{R}$ in such a way that the corresponding asignation \begin{align*} C_{f}(X)&\longrightarrow \mbox{LC}\big((\mbox{Ends}(X)\big)\\ g&\longmapsto \overline{g}|_{\mbox{Ends}(X)} \end{align*} is a canonical surjective homomorphism of real algebras whose kernel is $C_{c}(X)$. As LC($-$) is a contravariant equivalence between the category of compact Hausdorff totally disconnected spaces and continuous functions and the category of real algebras linearly generated by idempotents with algebra homomorphisms, it follows that two rimcompact Tychonoff spaces $X$ and $Y$ have the same end space (up to homeomorphism) if and only there is an algebra isomorphism $C_{f}(X)\rightarrow C_{f}(Y)$ carrying $C_{c}(X)$ to $C_{c}(Y)$.
What I would like to do is to state the last conclusion of the previous paragraph for proper geodesic metric spaces by using functions not necessarily continuous. More specifically, if we consider a proper geodesic space $X$ and the analogous real algebras \begin{align*} B_{f}(X)&=\{f: X\rightarrow \mathbb{R}\hspace{0.1cm}|\hspace{0.1cm}f \mbox{ is locally constant outside a compact subset}\},\\ B_{c}(X)&=\{f: X\rightarrow \mathbb{R}\hspace{0.1cm}|\hspace{0.1cm}f \mbox{ is zero outside a compact subset}\}, \end{align*} I want to prove that $B_{f}(X)=C_{f}(X)+B_{c}(X)$ and $C_{c}(X)=B_{c}(X)\cap C_{d}(X)$ in a canonical way. By the second isomorphism theorem for real algebras, the two previous equalities would imply that $$C_{f}(X)/C_{c}(X)\cong B_{f}(X)/B_{c}(X)$$ by a canonical isomorphism and thus I would get that two proper geodesic spaces $X$ and $Y$ have the same end space if and only if there is an algebra homomorphism $B_{f}(X)\rightarrow B_{f}(Y)$ carrying $B_{c}(X)$ to $B_{c}(Y)$.
Following Lemma 2.40 of John Roe's book Lectures on Coarse Geometry, I think the equality $B_{f}(X)=C_{f}(X)+B_{c}(X)$ can be archieved as follows. Consider a open cover $\{U_{\alpha}\}_{\alpha\in I}$ of $X$ consisting of open balls of equal radii and a locally finite partition of unity $\{\phi_{\alpha}\}_{\alpha\in I}$ subordinate to $\{U_{\alpha}\}$. Choosing a point $x_{\alpha}\in U_{\alpha}$, for every $g\in B_{f}(X)$ define the continuous function \begin{align} h:X&\longrightarrow \mathbb{R}\\ x&\longmapsto\sum_{\alpha\in I}\phi_{\alpha}(x)g(x_{\alpha}). \end{align} It should be the case that $h\in C_{f}(X)$ and $h-g\in B_{c}(X)$. Moreover, I suspect that if $K\subseteq X$ is a compact subset for which $g(X\setminus K)$ is finite, then $g(x)=h(x)$ for every $x\in X\setminus K$, but I have not been able to prove any of these two statements. One way I thougth I could proceed was to use properness and $\textit{geodesicness}$ to replace the compact set $K$ for a closed ball of radio $M>0$, say $\overline{B}(x_0,M)$, such that its complement $X\setminus \overline{B}(x_0,M)$ does not have bounded connected components and only has a finite number of unbounded connected components $\{U_{1},\cdots,U_{m}\}$. Considering those connected components along with the open ball $V=B(x_0,M+r)$, we get an open cover of $X$, $\{U_{1},\cdots,U_{m},V\}$, for which the associated function $h$ constructed above almost has the desired properties if the point $x_{V}$ is taken on $B(x_0,M+r)\setminus \overline{B}(x_0,r)$. The only problem is given by the points $x\not\in \overline{B}(x_0,M)$ for which the function $\phi_{V}$ whose support is contained in $B(x_0,M+r)$ is such that $\phi_{V}(x)\not=0$. In said situation one would like to prove that the point $x_{V}\in B(x_0,M+r)$ has to be in the same component of $x$ in order to conclude that $g(x)=g(x_{V})$, but I couldn't be able to verify it.
Actually, John Roe freely uses the fact that $B_{f}(X)=C_{f}(X)+B_{c}(X)$ and $C_{c}(X)=B_{c}(X)\cap C_{d}(X)$ on page 15 of this paper when $B_{f}(X)$ and $B_{c}(X)$ consist on Borel functions fulfilling the natural requirements, but he didn't give any reference or reason for that.
Any help, suggestion or reference is greately appreciated.
$\textbf{Edit}$:
Originally I defined the algebra $B_{f}(X)$ to consist of real-valued functions with finite image outside a compact set. The change to real-valued functions which are locally constant outside a compact set is neccesary to reconcile the equalities I want to prove with the equalities used by John Roe. For real-valued continuous functions on a proper geodesic space $X$, those algebras are equivalent, but even for Borel functions (the ones considered by John Roe) a priori can only be assured that a function which is locally constant outside a compact set $K$ is finite outside a (possibly different) compact set $C$. In general terms, the converse can fail because the image of a connected subset under a non-continuous function can be disconnected.