if A is positive semidefinite and B,C are positive definite matrices, can you bound the spectral norm $\lVert B(I+AB)^{-1}AC - B(I+AC)^{-1}AC\rVert $ by something like $\lVert B\rVert \lVert B - C\rVert \lVert C\rVert$?
So I guess the way to do it is $(I+AB)^{-1}- (I+AC)^{-1}=(I+AB)^{-1}(I+AB-I-AC)(I+AC)^{-1}$ and bound this ?