Purpose of Extended Kalman Filter with Constant Minimum MSE Matrices
Are there good reasons not to compute these matrices?

Yes.  The computation is expensive, particularly for large systems.  In addition, proving stability, particularly for an extended Kalman filter, can range from hard to impossible.
A KF where you don't update those matrices is a known thing; it's called a &quot;steady-state Kalman filter&quot; and it's based on the notion that for some systems $M_{n|n-1}$ and $M_{n|n}$ will asymptotically approach constant values.  To construct one, you compute* the values of $M_{n|n-1}$ and $M_{n|n}$ in the limit as $n \to \infty$, and use those rather than the continually-updated ones.
Depending on the system, a steady-state Kalman filter can be stable, well-behaved, and much less computationally expensive than a full-on Kalman.  The downside is that it takes longer for the errors to settle out, and in the case of an extended Kalman filter that asymptotic performance may not be as good as it could be because the filter doesn't match the system as well -- but for a lot of systems the provable stability and lower computational burden may be Very Good Things.
* You compute these either by just running the computation forward offline and saving the values, or by using the Ricatti equation to compute them directly.