($C', \mathfrak{R}$) is satisfiable in $\mathcal{ALC}_{\mathcal{R}\mathfrak{A}\ominus}$ iff the original concept $C$ is.$^2$
$C''$ is constructed from $C$ as follows: The role $\oplus(R)$ in $C$ is replaced by the role $R_{\oplus}$. Then, for every role $R_{\oplus}$, we add the role axioms ${R \circ R \subseteq R_{\oplus}, R_{\oplus} \circ R \subseteq R_{\oplus}}$ to $\mathfrak{R}$. Please note that this only ensures $(\oplus(R))^\mathcal{I} = R^\mathcal{I} \cup R_{\oplus}^\mathcal{I}$, and not $(\oplus(R))^\mathcal{I} = R_{\oplus}^\mathcal{I}$, since $R^\mathcal{I} \not\subseteq R_{\oplus}^\mathcal{I}$. Therefore, in order to get an equi-satisfiable concept $C'$, we have to rewrite the original concept $C$ in the following way:
Now, $C'$ is satisfiable w.r.t. the role box $\mathfrak{R}$ iff $C$ is satisfiable.
$\mathcal{ALCH}_{R+}$: The description logic $\mathcal{ALCH}{R+}$ (see [13, 14]) extends $\mathcal{ALC}{R+}$ by an additional set of role inclusion axioms of the form $R \subseteq S$, enforcing $R^\mathcal{I} \subseteq S^\mathcal{I}$ on the models $\mathcal{I}$. Adding the identity role $Id$ with the fixed semantics of the identity relationship $Id^\mathcal{I} ={def} {\langle x, x \rangle \mid x \in \Delta^\mathcal{I} }$ to $\mathcal{ALC}{RA\ominus}$ would obviously enable the simulation of these role inclusion axioms: for each role inclusion axiom $R \subseteq S$, add the role axiom $R \circ Id \subseteq S$ to a role box $\mathfrak{R}$ and consider the concept satisfiability w.r.t. $\mathfrak{R}$. Currently, neither $\mathcal{ALC}{RA\ominus}$ nor $\mathcal{ALC}{RA}$ provide the identity role.
Other Fragments of FOPL: In the following we will briefly discuss whether decidability or undecidability of $\mathcal{ALC}{RA\ominus}$ follows from already known results in logic, namely from results in bounded number of variables FOPL, or results from research carried out in the so-called (loosely) guarded fragment of FOPL. To the best of our knowledge, no previously known decidability resp. undecidability result is exploitable in the case of $\mathcal{ALC}{RA\ominus}$.
It is well-known that certain fragments of FOPL are decidable, for example, the class of all closed FOPL formulas containing at most two variables, denoted by $FO^2$. $FO^2$ has the finite model property – each satisfiable formula has a finite model. We already noted that one would need at least three variables if one translates $\mathcal{ALC}{RA\ominus}$ role boxes and concepts into FOPL. In fact, there is no way even to express the transitivity axiom $\forall x, y, z : R(x, y) \land R(y, z) \Rightarrow R(x, z)$ in $FO^2$ (see [10]). If $FO^2$ is augmented by transitivity on an extra-logical level (since transitivity cannot be expressed within the language itself), $FO^2$ becomes undecidable, as Grädel et al. have shown (see [10]). However, the class $FO^2$ is much too large to capture the concept-side of $\mathcal{ALC}{RA\ominus}$, since $\mathcal{ALC}$ concepts are expressible in a proper subset of $FO^2$, namely $GF_\beta^2$, see below. Recall that $\mathcal{ALC}_{R+}$ is decidable.
$^2$It follows that if $\mathcal{ALC}_{RA\ominus}$ was decidable it would be EXPTIME-hard, since $\mathcal{ALC}_\oplus$ is EXPTIME-complete.