Personally I prefer $\hat\sigma^2$ over $\hat\sigma_{MMSE}^2$ for a reason different from unbiasedness. If the estimation problem were in fact symmetric, i.e., too low values would be as bad as too high values of the same size, I'd think that the MSE would be a good measure, and optimum MSE would be in fact good. (Note that also the expected value, and by extension the concept of unbiasedness, implicitly treats the estimation problem as symmetric, as deviations on both sides count the same also for the computation of the expected value.)
But actually variances are bounded from below, and differences between small variances should be taken as more important as the same differences between large variances; also, by implication, there should be more loss for the same deviation in negative direction than in positive direction (if the true $\sigma^2$ is 1, 0.5 should be seen as worse than 1.5; in fact, in robust statistics, $\hat\sigma^2\to 0$ is seen as "breakdown" along with $\hat\sigma^2\to \infty$).
So in fact one could think that we'd need an estimator to optimise an asymmetric loss function rather than MSE, and correspondingly there would even need to be an alternative definition to unbiasedness. Now this would be hard to do; as it isn't at all obvious how this loss function should look like, and different choices would have different implications. One could probably do some research work on this and publish a nice paper, but in practical analysis situations, that's not really what we want to do.
So in standard routine data analysis I bite the bullet and use $\hat\sigma^2$ as this is implemented everywhere and I can explain its features in a fairly generally understandable way, even though secretly I think that, because of the unbiasedness feature ignoring asymmetry, $\hat\sigma^2$ likely tends to be too small.
I wouldn't worry much about any "complication" as a result of using $\hat\sigma_{MMSE}^2$; just using a different factor would be easy to do, and the optimum MSE argument is appealing on the surface, but in fact $\hat\sigma_{MMSE}^2$ is even smaller than $\hat\sigma^2$, which I honestly think is rather too small already, if anything.
So no, $\hat\sigma_{MMSE}^2$ is not a good alternative!
PS: Of course one could have the same discussion involving the Maximum Likelihood variance estimator with factor $\frac{1}{n}$, which is sometimes used, and has a better MSE than $\hat\sigma^2$.
PPS: One could actually interpret the difference between the optimum unbiased, the ML, and the minimum MSE estimator as an expression of the asymmetry of the problem.