Suppose I have a matrix equation of the form: $$ C = ((A+X)^{-1}+B)^{-1}\tag{1} $$ where $A$, $B$, and $C$ are all squares and I want to solve for the unknown square $X$. (All of them are of the same dimension, $n$-by-$n$.)
Now, assume such an $X$ exists. Then I see a number of ways to proceed. First, I can left-multiply both sides of (1) by $(A+X)^{-1}+B$, getting $((A+X)^{-1}+B)C = I$, where $I$ is the identity matrix of the appropriate size. Left-multiplying both sides by $A+X$ yields $(I + (A+X)B)C = A +X$, which after rearranging gives an expression of $$X = (C - A + ABC)(I - BC)^{-1},\tag{2}$$ assuming $I - BC$ is invertible.
Second, I can instead right-multiply both sides of (1) by $(A+X)^{-1}+B$, getting $C((A+X)^{-1}+B)=I$. Further right-multiplying both sides by $A+X$ yields $C(I+B(A+X))=A+X$, which gives another expression of $$X = (I - CB)^{-1}(C - A + CBA),\tag{3}$$ assuming $I - CB$ is invertible.
Yet another approach is to let $Z=(A+X)^{-1}$, converting (1) into $C = (Z + B)^{-1}$, which yields $Z = C^{-1} - B$, assuming $C$ is invertible. Then it follows that $$X = (C^{-1} - B)^{-1} - A\tag{4}.$$
So it seems that I have recovered three different solutions of the original problem of (1). In particular, the solutions (2)-(4) all seem to be necessary outcomes following (1), assuming the invertibility of $I-BC$, of $I-CB$, and of $C$.
However, in the numerical experiments I've tried, I can always reject (2). For example, $$ A = \begin{bmatrix}1&2\\0&3\end{bmatrix},\; B = \begin{bmatrix}0&3\\-1&2\end{bmatrix},\;\text{and}\; C = \begin{bmatrix}1&0\\0&1\end{bmatrix} $$ which satisfy the invertibility assumptions. In this example, both (3) and (4) yield the same $X = \begin{bmatrix}-1.5&-0.5\\-0.5&-2.5\end{bmatrix}$, indeed solving (1), but the solution implied by (2) fails.
My question, in summary, is: In which steps of the above did I mess up with the necessity, especially in deriving the expression (2)? (This is particularly curious to me as the difference between (2) and (3) is only the left-multiplication vs. the right-multiplication.)
To clarify, I am not explicitly asking for all the solutions to (1). I simply want to understand what went wrong above in deriving (2). But of course, it would be nice to know further: why (3) and (4) are equivalent numerically, even though they seem very different; and whether (3) or (4) is the unique solution to (1).
