Given two matrices $A \in \mathbb{R^{m \times k}}, B \in \mathbb{R^{n \times k}}$ with $k> m > n$, I want to compute $(A B^{\dagger})^{\dagger} A$ where $\cdot^{\dagger}$ is the Moore-Penrose inverse. If it were true that $(A B^{\dagger})^{\dagger} A = B A^{\dagger} A$ (which I know is not in my case), I could crucially accelerate an algorithm of mine (because $B$ changes with each iteration but $A$ is fixed, so $A^{\dagger} A$ could be precomputed).
I was wondering, if there exists a good approximation of the error $(A B^{\dagger})^{\dagger} A - B A^{\dagger} A$ that does not rely on computing $B^{\dagger}$. Or alternatively, if there exists a good approximation of $(A B^{\dagger})^{\dagger} A$ without computing $B^{\dagger}$.
Such an approximation might not exist and maybe my question is complete nonsense but I thought, maybe you have some ideas! Thanks.